Device and method of controlling exhaust gas sensor temperature, and recording medium for exhaust gas sensor temperature control program

ABSTRACT

A control input (DUT) for controlling a heater ( 13 ) which heats an active element ( 10 ) of an exhaust gas sensor ( 8 ) includes at least one of another component depending on the difference between temperature data of the active element ( 10 ) and a target temperature, a component depending on the target temperature, and a component depending on the temperature data of the active element ( 10 ). The control input is determined by an optimum control algorithm. A component depending on the temperature of an exhaust gas and the component depending on the target temperature are determined based on a predictive control algorithm. The temperature of the active element ( 10 ) of the exhaust gas sensor ( 8 ) is thus controlled stably at a desired temperature.

TECHNICAL FIELD

The present invention relates to an apparatus for and a method ofcontrolling the temperature of an exhaust gas sensor disposed in theexhaust passage of an internal combustion engine, and a recording mediumstoring a program for controlling the temperature of such an exhaust gassensor.

BACKGROUND ART

Exhaust gas sensors are often disposed in the exhaust passages ofinternal combustion engines for detecting a physical quantity as to anexhaust gas component state, such as an exhaust gas componentconcentration, for the purpose of controlling the operation of theinternal combustion engine or monitoring the status of an exhaust gaspurifying system. Specifically, the exhaust gas sensor is disposed at acertain location in the exhaust gas passage and has an element sensitiveto an exhaust gas component state to be detected, the element beingpositioned for contact with an exhaust gas flowing through the exhaustpassage. For example, an air-fuel ratio sensor such as an O₂ sensor orthe like is disposed as an exhaust gas sensor upstream or downstream ofan exhaust gas purifying catalyst disposed in the exhaust passage forthe purpose of controlling the air-fuel ratio of the internal combustionengine in order to keep well the purifying ability of the catalyst.

Some air-fuel ratio sensors have a built-in heater for heating theactive element thereof for increasing the temperature of the element andactivating the element to enable the element to perform its essentialfunctions and also removing foreign matter deposited on the element. Forexample, an air-fuel ratio sensor such as an O₂ sensor or the likeusually has an electric heater for heating the active element thereof.After the internal combustion engine has started to operate, theelectric heater is energized to increase the temperature of the activeelement of the O₂ sensor to activate the active element and keep theactive element active.

As shown in FIG. 3 of the accompanying drawings, the O₂ sensor producesan output voltage Vout which changes with a large gradient with respectto a change in the air-fuel ratio of an exhaust gas, i.e., which ishighly sensitive to a change in the air-fuel ratio, only in a smallrange Δ (near a stoichiometric air-fuel ratio) of values of the air-fuelratio that is represented by an oxygen concentration in the exhaust gasto which the active element is sensitive. A change in the output voltageVout of the O₂ sensor, i.e., a gradient of the output voltage Vout withrespect to the air-fuel ratio, is smaller in air-fuel ratio ranges thatare richer and leaner than the highly sensitive range Δ. The outputcharacteristics of the O₂ sensor, i.e., the gradient of the highlysensitive range Δ, etc., vary depending on the temperature of the activeelement. When the air-fuel ratio is to be controlled using the outputvoltage from the O₂ sensor, therefore, it is desirable to keep theoutput characteristics of the O₂ sensor in a desired range as much aspossible and hence to keep the temperature of the active element of theO₂ sensor in a desired temperature range as stably as possible forbetter air-fuel ratio control.

Not only O₂ sensors but also many exhaust gas sensors have their outputcharacteristics affected by the temperature of the active element. Ifthe internal combustion engine is to be controlled using the outputsignal from the O₂ sensor, then it is preferable to keep the temperatureof the active element of the exhaust gas sensor in a desired temperaturerange as stably as possible for better engine control. When the activeelement of the exhaust gas sensor is heated to clean the active element,it is also preferable to keep the temperature of the active element ofthe exhaust gas sensor in a desired temperature range for a bettercleaning effect.

As disclosed in Japanese laid-open patent publication No. 2000-304721 bythe applicant of the present application, it is known to estimate thetemperature of the active element of an exhaust gas sensor (an air-fuelratio sensor in the publication) and control the energization of aheater (an electric heater) based on the estimated temperature forthereby keep the temperature of the active element in a desiredtemperature range to obtain appropriate output characteristics from theexhaust gas sensor. According to the disclosed arrangement, theresistance of the heater is recognized from detected values of a currentflowing through the heater and a voltage applied across the heater, andthe temperature of the active element is estimated based on the detectedresistance of the heater.

According to the disclosure of the above publication, however, since aduty cycle which determines the electric power to be supplied to theheater is uniquely determined by a table from an estimated value of thetemperature of the active element of the exhaust gas sensor, it isdifficult to control the temperature of the active element of theexhaust gas sensor stably at a desired temperature because of a changein the temperature of the exhaust gas, an ambient air temperature, etc.Furthermore, if the temperature of the active element of the exhaust gassensor is low, then in order to keep the duty cycle of the heater at acertain maximum level, the heater tends to consume more electric powerthan required and the actual temperatures of the heater and the activeelement are liable to be excessively high, causing damage to the heaterand the active element.

The present invention has been made in view of the above background. Itis an object of the present invention to provide an apparatus for and amethod of controlling the temperature of the active element of anexhaust gas sensor stably at a desired temperature. Another object ofthe present invention is to provide a recording medium storing atemperature control program for controlling the temperature of theactive element of an exhaust gas sensor stably at a desired temperature.

DISCLOSURE OF THE INVENTION

An apparatus for controlling the temperature of an exhaust gas sensoraccording to the present invention is an apparatus for controlling thetemperature of an exhaust gas sensor disposed in an exhaust passage ofan internal combustion engine and having an active element forcontacting an exhaust gas flowing through the exhaust passage and aheater for heating the active element. To achieve the above object,according to a first aspect of the present invention, an apparatus forcontrolling the temperature of an exhaust gas sensor is characterized bycomprising means for sequentially acquiring element temperature datarepresenting the temperature of the active element, means forsequentially acquiring heater temperature data representing thetemperature of the heater, and heater control means for sequentiallygenerating a control input which defines an amount of heat generatingenergy supplied to the heater so as to equalize the temperature of theactive element represented by the element temperature data to apredetermined target temperature, and controlling the heater dependingon the control input, and characterized in that the control inputgenerated by the heater control means includes at least an inputcomponent depending on the difference between the temperature of theactive element represented by the element temperature data and thetarget temperature and an input component depending on the temperatureof the heater represented by the heater temperature data.

Similarly, a method of controlling the temperature of an exhaust gassensor according to the present invention is a method of controlling thetemperature of an exhaust gas sensor disposed in an exhaust passage ofan internal combustion engine and having an active element forcontacting an exhaust gas flowing through the exhaust passage and aheater for heating the active element. According to the first aspect ofthe present invention, a method of controlling the temperature of anexhaust gas sensor is characterized by comprising the steps ofsequentially acquiring element temperature data representing thetemperature of the active element and heater temperature datarepresenting the temperature of the heater, sequentially generating acontrol input which defines an amount of heat generating energy suppliedto the heater so as to equalize the temperature of the active elementrepresented by the element temperature data to a predetermined targettemperature, and controlling the heater depending on the control input,and characterized in that when the control input is generated, thecontrol input is generated so as to include at least an input componentdepending on the difference between the temperature of the activeelement represented by the element temperature data and the targettemperature and an input component depending on the temperature of theheater represented by the heater temperature data.

Furthermore, a recording medium storing a temperature control programfor an exhaust gas sensor according to the present invention is arecording medium readable by a computer and storing a temperaturecontrol program for enabling the computer to perform a process ofcontrolling the temperature o an active element of an exhaust gas sensordisposed in an exhaust passage of an internal combustion engine andhaving the active element for contacting an exhaust gas flowing throughthe exhaust passage and a heater for heating the active element.According to the first aspect of the present invention, a recordingmedium storing a temperature control program for an exhaust gas sensoris characterized in that the temperature control program includes aprogram for enabling the computer to perform a process of sequentiallyacquiring element temperature data representing the temperature of theactive element and heater temperature data representing the temperatureof the heater, a control input generating program for enabling thecomputer to perform a process of sequentially generating a control inputwhich defines an amount of heat generating energy supplied to the heaterso as to equalize the temperature of the active element represented bythe element temperature data to a predetermined target temperature, anda program for enabling the computer to perform a process of controllingthe heater depending on the control input, wherein the control inputgenerating program has an algorithm for enabling the computer togenerate the control input so as to include at least an input componentdepending on the difference between the temperature of the activeelement represented by the element temperature data and the targettemperature and an input component depending on the temperature of theheater represented by the heater temperature data. According to thefirst aspect of the present invention, since the heater is controlledbased on a control input (a manipulated variable for an object to becontrolled) including an input component depending on the differencebetween the temperature of the active element represented by the elementtemperature data and the target temperature and an input componentdepending on the temperature of the heater represented by the heatertemperature data, or stated otherwise, a control input generated bycombining at least the above input components, when the temperature ofthe active element varies with respect to the target temperature, it ispossible to converge the temperature of the active element to the targettemperature while suppressing excessive variations of the control inputwhich defines an amount of heat generating energy supplied to theheater. As a result, the temperature of the active element of theexhaust gas sensor can stably be controlled at the target temperature.

According to the first aspect of the present invention, the elementtemperature data representative of the temperature of the active elementmay directly be detected and acquired using a temperature sensor, or maybe estimated and acquired based on a suitable parameter or a model. Theheater temperature data representative of the temperature of the heatermay also be acquired in the same manner.

The input component depending on the difference between the temperatureof the active element and the target temperature in the control inputmay be a component proportional to the difference, a componentproportional to an accumulative sum (integrated value) of values of thedifference at respective times, or a sum of these components. This alsoapplies to a second aspect, a third aspect, and a fourth aspect of thepresent invention which will be described later.

According to a second aspect of the present invention, an apparatus forcontrolling the temperature of an exhaust gas sensor is characterized bycomprising means for sequentially acquiring element temperature datarepresenting the temperature of the active element, means forsequentially acquiring exhaust gas temperature data representing thetemperature of the exhaust gas, and heater control means forsequentially generating a control input which defines an amount of heatgenerating energy supplied to the heater so as to equalize thetemperature of the active element represented by the element temperaturedata to a predetermined target temperature, and controlling the heaterdepending on the control input, and characterized in that the controlinput generated by the heater control means includes at least an inputcomponent depending on the difference between the temperature of theactive element represented by the element temperature data and thetarget temperature and an input component depending on the temperatureof the exhaust gas represented by the exhaust gas temperature data.

Similarly, according to the second aspect of the present invention, amethod of controlling the temperature of an exhaust gas sensor ischaracterized by comprising the steps of sequentially acquiring elementtemperature data representing the temperature of the active element andexhaust gas temperature data representing the temperature of the exhaustgas, sequentially generating a control input which defines an amount ofheat generating energy supplied to the heater so as to equalize thetemperature of the active element represented by the element temperaturedata to a predetermined target temperature, and controlling the heaterdepending on the control input, and characterized in that when thecontrol input is generated, the control input is generated so as toinclude at least an input component depending on the difference betweenthe temperature of the active element represented by the elementtemperature data and the target temperature and an input componentdepending on the temperature of the exhaust gas represented by theexhaust gas temperature data.

Furthermore, according to the second aspect of the present invention, arecording medium storing a temperature control program for an exhaustgas sensor is characterized in that the temperature control programincludes a program for enabling the computer to perform a process ofsequentially acquiring element temperature data representing thetemperature of the active element and exhaust gas temperature datarepresenting the temperature of the exhaust gas, a control inputgenerating program for enabling the computer to perform a process ofsequentially generating a control input which defines an amount of heatgenerating energy supplied to the heater so as to equalize thetemperature of the active element represented by the element temperaturedata to a predetermined target temperature, and a program for enablingthe computer to perform a process of controlling the heater depending onthe control input, wherein the control input generating program has analgorithm for enabling the computer to generate the control input so asto include at least an input component depending on the differencebetween the temperature of the active element represented by the elementtemperature data and the target temperature and an input componentdepending on the temperature of the exhaust gas represented by theexhaust gas temperature data.

According to the second aspect of the present invention, since theheater is controlled based on a control input including an inputcomponent depending on the difference between the temperature of theactive element represented by the element temperature data and thetarget temperature and an input component depending on the temperatureof the exhaust gas represented by the exhaust gas temperature data, orstated otherwise, a control input generated by combining at least theabove input components, it is possible to converge the temperature ofthe active element to the target temperature while compensating forvariations of the temperature of the exhaust gas with respect to thetemperature of the active element. Stated otherwise, it is possible toconverge the temperature of the active element to the target temperaturewhile suppressing variations of the temperature of the active elementdue to variations of the temperature of the exhaust gas. As a result,the temperature of the active element of the exhaust gas sensor canstably be controlled at the target temperature (desired temperature).

According to the second aspect of the present invention, the elementtemperature data representative of the temperature of the active elementmay directly be detected and acquired using a temperature sensor, or maybe estimated and acquired based on a suitable parameter or a model, aswith the first aspect. The exhaust gas temperature data representativeof the temperature of the exhaust gas may also be acquired in the samemanner.

According to a third aspect of the present invention, an apparatus forcontrolling the temperature of an exhaust gas sensor is characterized bycomprising means for sequentially acquiring element temperature datarepresenting the temperature of the active element, and heater controlmeans for sequentially generating a control input which defines anamount of heat generating energy supplied to the heater so as toequalize the temperature of the active element represented by theelement temperature data to a predetermined target temperature, andcontrolling the heater depending on the control input, and characterizedin that the control input generated by the heater control means includesat least an input component depending on the difference between thetemperature of the active element represented by the element temperaturedata and the target temperature and an input component depending on thetarget temperature.

Similarly, according to the third aspect of the present invention, amethod of controlling the temperature of an exhaust gas sensor ischaracterized by comprising the steps of sequentially acquiring elementtemperature data representing the temperature of the active element,sequentially generating a control input which defines an amount of heatgenerating energy supplied to the heater so as to equalize thetemperature of the active element represented by the element temperaturedata to a predetermined target temperature, and controlling the heaterdepending on the control input, and characterized in that when thecontrol input is generated, the control input is generated so as tofurther include at least an input component depending on the differencebetween the temperature of the active element represented by the elementtemperature data and the target temperature and an input componentdepending on the target temperature.

Furthermore, according to the third aspect of the present invention, arecording medium storing a temperature control program for an exhaustgas sensor is characterized in that the temperature control programincludes a program for enabling the computer to perform a process ofsequentially acquiring element temperature data representing thetemperature of the active element, a control input generating programfor enabling the computer to perform a process of sequentiallygenerating a control input which defines an amount of heat generatingenergy supplied to the heater so as to equalize the temperature of theactive element represented by the element temperature data to apredetermined target temperature, and a program for enabling thecomputer to perform a process of controlling the heater depending on thecontrol input, wherein the control input generating program has analgorithm for enabling the computer to generate the control input so asto include at least an input component depending on the differencebetween the temperature of the active element represented by the elementtemperature data and the target temperature and an input componentdepending on the target temperature.

According to the third aspect of the present invention, since the heateris controlled based on a control input including an input componentdepending on the difference between the temperature of the activeelement represented by the element temperature data and the targettemperature and an input component depending on the target temperature,or stated otherwise, a control input generated by combining at least theabove input components, it is possible to converge the temperature ofthe active element quickly to the target temperature. As a result, thetemperature of the active element of the exhaust gas sensor can stablybe controlled at the target temperature.

According to the third aspect of the present invention, the elementtemperature data representative of the temperature of the active elementmay directly be detected and acquired using a temperature sensor, or maybe estimated and acquired based on a suitable parameter or a model, aswith the first aspect.

According to a fourth aspect of the present invention, an apparatus forcontrolling the temperature of an exhaust gas sensor is characterized bycomprising means for sequentially acquiring element temperature datarepresenting the temperature of the active element, and heater controlmeans for sequentially generating a control input which defines anamount of heat generating energy supplied to the heater so as toequalize the temperature of the active element represented by theelement temperature data to a predetermined target temperature, andcontrolling the heater depending on the control input, and characterizedin that the control input generated by the heater control means includesat least an input component depending on the difference between thetemperature of the active element represented by the element temperaturedata and the target temperature and an input component depending on thetemperature of the active element.

Similarly, according to the fourth aspect of the present invention, amethod of controlling the temperature of an exhaust gas sensor ischaracterized by comprising the steps of sequentially acquiring elementtemperature data representing the temperature of the active element,sequentially generating a control input which defines an amount of heatgenerating energy supplied to the heater so as to equalize thetemperature of the active element represented by the element temperaturedata to a predetermined target temperature, and controlling the heaterdepending on the control input, and characterized in that when thecontrol input is generated, the control input is generated so as toinclude at least an input component depending on the difference betweenthe temperature of the active element represented by the elementtemperature data and the target temperature and an input componentdepending on the temperature of the active element.

Furthermore, according to the fourth aspect of the present invention, arecording medium storing a temperature control program for an exhaustgas sensor is characterized in that the temperature control programincludes a program for enabling the computer to perform a process ofsequentially acquiring element temperature data representing thetemperature of the active element, a control input generating programfor enabling the computer to perform a process of sequentiallygenerating a control input which defines an amount of heat generatingenergy supplied to the heater so as to equalize the temperature of theactive element represented by the element temperature data to apredetermined target temperature, and a program for enabling thecomputer to perform a process of controlling the heater depending on thecontrol input, wherein the control input generating program has analgorithm for enabling the computer to generate the control input so asto include at least an input component depending on the differencebetween the temperature of the active element represented by the elementtemperature data and the target temperature and an input componentdepending on the temperature of the active element.

According to the fourth aspect of the present invention, since theheater is controlled based on a control input including an inputcomponent depending on the difference between the temperature of theactive element represented by the element temperature data and thetarget temperature and an input component depending on the targettemperature for the active element, or stated otherwise, a control inputgenerated by combining at least the above input components, it ispossible to reduce overshooting of the temperature of the active elementwith respect to the target temperature, and to allow the temperature ofthe active element to smoothly follow the target temperature. As aresult, the temperature of the active element of the exhaust gas sensorcan stably be controlled at the target temperature.

According to the fourth aspect of the present invention, the elementtemperature data representative of the temperature of the active elementmay directly be detected and acquired using a temperature sensor, or maybe estimated and acquired based on a suitable parameter or a model, aswith the first aspect.

In the first through fourth aspects of the invention, the temperature ofthe active element is directly controlled at the target temperature.Generally, the temperature of the heater and the temperature of theactive element are highly correlated to each other in a steady statewhere those temperatures are substantially constant. Therefore, when thetemperature of the heater is controlled at a certain target temperature,the temperature of the active element can indirectly be controlled at atemperature corresponding to the target temperature for the heater.

In view of the foregoing, according to a fifth aspect of the presentinvention, an apparatus for controlling the temperature of an exhaustgas sensor is characterized by comprising means for sequentiallyacquiring element temperature data representing the temperature of theactive element, means for sequentially acquiring heater temperature datarepresenting the temperature of the heater, and heater control means forsequentially generating a control input which defines an amount of heatgenerating energy supplied to the heater so as to equalize thetemperature of the heater represented by the heater temperature data toa predetermined target temperature, and controlling the heater dependingon the control input, and characterized in that the control inputgenerated by the heater control means includes at least an inputcomponent depending on the difference between the temperature of theheater represented by the heater temperature data and the targettemperature, and an input component depending on the temperature of theactive element represented by the element temperature data.

Similarly, according to the fifth aspect of the present invention, amethod of controlling the temperature of an exhaust gas sensor ischaracterized by comprising the steps of sequentially acquiring elementtemperature data representing the temperature of the active element andheater temperature data representing the temperature of the heater,sequentially generating a control input which defines an amount of heatgenerating energy supplied to the heater so as to equalize thetemperature of the heater represented by the heater temperature data toa predetermined target temperature, and controlling the heater dependingon the control input, and characterized in that when the control inputis generated, the control input is generated so as to include at leastan input component depending on the difference between the temperatureof the heater represented by the heater temperature data and the targettemperature, and an input component depending on the temperature of theactive element represented by the element temperature data.

Furthermore, according to the fifth aspect of the present invention, arecording medium storing a temperature control program for an exhaustgas sensor is characterized in that the temperature control programincludes a program for enabling the computer to perform a process ofsequentially acquiring element temperature data representing thetemperature of the active element and heater temperature datarepresenting the temperature of the heater, a control input generatingprogram for enabling the computer to perform a process of sequentiallygenerating a control input which defines an amount of heat generatingenergy supplied to the heater so as to equalize the temperature of theheater represented by the heater temperature data to a predeterminedtarget temperature, and a program for enabling the computer to perform aprocess of controlling the heater depending on the control input,wherein the control input generating program has an algorithm forenabling the computer to generate the control input so as to include atleast an input component depending on the difference between thetemperature of the heater represented by the heater temperature data andthe target temperature, and an input component depending on thetemperature of the active element represented by the element temperaturedata.

The fifth aspect of the invention corresponds to the first aspect of theinvention, and provides advantages similar to those of the first aspectin relation to controlling the temperature of the heater at the targettemperature. Specifically, according to the fifth aspect of theinvention, since the heater is controlled based on a control input (amanipulated variable for an object to be controlled) including an inputcomponent depending on the difference between the temperature of theheater represented by the heater temperature data and the targettemperature (a target temperature for the heater) and an input componentdepending on the temperature of the active element represented by theelement temperature data, or stated otherwise, a control input generatedby combining at least the above input components, when the temperatureof the heater varies with respect to the target temperature, it ispossible to converge the temperature of the heater to the targettemperature while suppressing excessive variations of the control input.As a result, the temperature of the heater of the exhaust gas sensor canstably be controlled at the target temperature, and hence thetemperature of the active element can stably be controlled at atemperature corresponding to the target temperature for the heater.

According to the fifth aspect of the present invention, the elementtemperature data representative of the temperature of the active elementand the heater temperature data representative of the temperature of theheater may be detected and acquired by a temperature sensor, or may beestimated and acquired, as with the first aspect.

The input component depending on the difference between the temperatureof the heater and the target temperature in the control input may be acomponent proportional to the difference, a component proportional to anaccumulative sum (integrated value) of values of the difference atrespective times, or a sum of these components. This also applies to asixth aspect, a seventh aspect, and an eighth aspect of the presentinvention which will be described later.

According to a sixth aspect of the present invention, an apparatus forcontrolling the temperature of an exhaust gas sensor is characterized bycomprising means for sequentially acquiring heater temperature datarepresenting the temperature of the heater, means for sequentiallyacquiring exhaust gas temperature data representing the temperature ofthe exhaust gas, and heater control means for sequentially generating acontrol input which defines an amount of heat generating energy suppliedto the heater so as to equalize the temperature of the heaterrepresented by the heater temperature data to a predetermined targettemperature, and controlling the heater depending on the control input,and characterized in that the control input generated by the heatercontrol means includes at least an input component depending on thedifference between the temperature of the heater represented by theheater temperature data and the target temperature, and an inputcomponent depending on the temperature of the temperature of the exhaustgas represented by the exhaust gas temperature data.

Similarly, according to the sixth aspect of the present invention, amethod of controlling the temperature of an exhaust gas sensor ischaracterized by comprising the steps of sequentially acquiring heatertemperature data representing the temperature of the heater and exhaustgas temperature data representing the temperature of the exhaust gas,sequentially generating a control input which defines an amount of heatgenerating energy supplied to the heater so as to equalize thetemperature of the heater represented by the heater temperature data toa predetermined target temperature, and controlling the heater dependingon the control input, and characterized in that when the control inputis generated, the control input is generated so as to include at leastan input component depending on the difference between the temperatureof the heater represented by the heater temperature data and the targettemperature, and an input component depending on the temperature of thetemperature of the exhaust gas represented by the exhaust gastemperature data.

Furthermore, according to the sixth aspect of the present invention, arecording medium storing a temperature control program for an exhaustgas sensor is characterized in that the temperature control programincludes a program for enabling the computer to perform a process ofsequentially acquiring heater temperature data representing thetemperature of the heater and exhaust gas temperature data representingthe temperature of the exhaust gas, a control input generating programfor enabling the computer to perform a process of sequentiallygenerating a control input which defines an amount of heat generatingenergy supplied to the heater so as to equalize the temperature of theheater represented by the heater temperature data to a predeterminedtarget temperature, and a program for enabling the computer to perform aprocess of controlling the heater depending on the control input,wherein the control input generating program has an algorithm forenabling the computer to generate the control input so as to include atleast an input component depending on the difference between thetemperature of the heater represented by the heater temperature data andthe target temperature and an input component depending on thetemperature of the exhaust gas represented by the exhaust gastemperature data.

The sixth aspect of the invention corresponds to the second aspect ofthe invention, and provides advantages similar to those of the secondaspect in relation to controlling the temperature of the heater at thetarget temperature. Specifically, according to the sixth aspect of theinvention, since the heater is controlled based on a control inputincluding an input component depending on the difference between thetemperature of the heater represented by the heater temperature data andthe target temperature and an input component depending on thetemperature of the exhaust gas represented by the exhaust gastemperature data, or stated otherwise, a control input generated bycombining at least the above input components, it is possible toconverge the temperature of the heater to the target temperature whilecompensating for variations of the temperature of the exhaust gas withrespect to the temperature of the active element and the temperature ofthe heater. Stated otherwise, it is possible to control the temperatureof the heater to the target temperature while suppressing variations ofthe temperature of the heater due to variations of the temperature ofthe exhaust gas. As a result, the temperature of the heater of theexhaust gas sensor can stably be controlled at the target temperature,and the temperature of the active element of the exhaust gas sensor canstably be controlled at a temperature corresponding to the targettemperature for the heater.

According to the sixth aspect of the present invention, the heatertemperature data representative of the temperature of the heater and theexhaust gas temperature data representative of the temperature of theexhaust gas may be detected and acquired or estimated and acquired, aswith the first aspect.

According to a seventh aspect of the present invention, an apparatus forcontrolling the temperature of an exhaust gas sensor is characterized bycomprising means for sequentially acquiring heater temperature datarepresenting the temperature of the heater, and heater control means forsequentially generating a control input which defines an amount of heatgenerating energy supplied to the heater so as to equalize thetemperature of the heater represented by the heater temperature data toa predetermined target temperature, and controlling the heater dependingon the control input, and characterized in that the control inputgenerated by the heater control means includes at least an inputcomponent depending on the difference between the temperature of theheater represented by the heater temperature data and the targettemperature and an input component depending on the target temperature.

Similarly, according to the seventh aspect of the present invention, amethod of controlling the temperature of an exhaust gas sensor ischaracterized by comprising the steps of sequentially acquiring heatertemperature data representing the temperature of the heater,sequentially generating a control input which defines an amount of heatgenerating energy supplied to the heater so as to equalize thetemperature of the heater represented by the heater temperature data toa predetermined target temperature, and controlling the heater dependingon the control input, and characterized in that when the control inputis generated, the control input is generated so as to include at leastan input component depending on the difference between the temperatureof the heater represented by the heater temperature data and the targettemperature and an input component depending on the target temperature.

Furthermore, according to the seventh aspect of the present invention, arecording medium storing a temperature control program for an exhaustgas sensor is characterized in that the temperature control programincludes a program for enabling the computer to perform a process ofsequentially acquiring heater temperature data representing thetemperature of the heater, a control input generating program forenabling the computer to perform a process of sequentially generating acontrol input which defines an amount of heat generating energy suppliedto the heater so as to equalize the temperature of the heaterrepresented by the heater temperature data to a predetermined targettemperature, and a program for enabling the computer to perform aprocess of controlling the heater depending on the control input,wherein the control input generating program has an algorithm forenabling the computer to generate the control input so as to include atleast an input component depending on the difference between thetemperature of the heater represented by the heater temperature data andthe target temperature and an input component depending on the targettemperature.

The seventh aspect of the invention corresponds to the third aspect ofthe invention, and provides advantages similar to those of the thirdaspect in relation to controlling the temperature of the heater at thetarget temperature. Specifically, according to the seventh aspect of theinvention, since the heater is controlled based on a control inputincluding an input component depending on the difference between thetemperature of the heater represented by the heater temperature data andthe target temperature and an input component depending on the targettemperature, or stated otherwise, a control input generated by combiningat least the above input components, it is possible to converge thetemperature of the heater quickly to the target temperature. As aresult, the temperature of the heater of the exhaust gas sensor canstably be controlled at the target temperature and the temperature ofthe active element can stably be controlled at a temperaturecorresponding to the target temperature for the heater.

According to the seventh aspect of the present invention, the heatertemperature data representative of the temperature of the heater may bedetected by a temperature sensor and acquired or may be estimated andacquired, as with the third aspect.

According to an eighth aspect of the present invention, an apparatus forcontrolling the temperature of an exhaust gas sensor is characterized bycomprising means for sequentially acquiring heater temperature datarepresenting the temperature of the heater, and heater control means forsequentially generating a control input which defines an amount of heatgenerating energy supplied to the heater so as to equalize thetemperature of the heater represented by the heater temperature data toa predetermined target temperature, and controlling the heater dependingon the control input, and characterized in that the control inputgenerated by the heater control means includes at least an inputcomponent depending on the difference between the temperature of theheater represented by the heater temperature data and the targettemperature and an input component depending on the temperature of theheater.

Similarly, according to the eighth aspect of the present invention, amethod of controlling the temperature of an exhaust gas sensor ischaracterized by comprising the steps of sequentially acquiring heatertemperature data representing the temperature of the heater,sequentially generating a control input which defines an amount of heatgenerating energy supplied to the heater so as to equalize thetemperature of the heater represented by the heater temperature data toa predetermined target temperature, and controlling the heater dependingon the control input, and characterized in that when the control inputis generated, the control input is generated so as to include at leastan input component depending on the difference between the temperatureof the heater represented by the heater temperature data and the targettemperature and an input component depending on the temperature of theheater.

Furthermore, according to the eighth aspect of the present invention, arecording medium storing a temperature control program for an exhaustgas sensor is characterized in that the temperature control programincludes a program for enabling the computer to perform a process ofsequentially acquiring heater temperature data representing thetemperature of the heater, a control input generating program forenabling the computer to perform a process of sequentially generating acontrol input which defines an amount of heat generating energy suppliedto the heater so as to equalize the temperature of the heaterrepresented by the heater temperature data to a predetermined targettemperature, and a program for enabling the computer to perform aprocess of controlling the heater depending on the control input,wherein the control input generating program has an algorithm forenabling the computer to generate the control input so as to include atleast an input component depending on the difference between thetemperature of the heater represented by the heater temperature data andthe target temperature and an input component depending on thetemperature of the heater.

The eighth aspect of the invention corresponds to the fourth aspect ofthe invention, and provides advantages similar to those of the fourthaspect in relation to controlling the temperature of the heater at thetarget temperature. Specifically, according to the eighth aspect of theinvention, since the heater is controlled based on a control inputincluding an input component depending on the difference between thetemperature of the heater represented by the heater temperature data andthe target temperature and an input component depending on thetemperature of the heater, or stated otherwise, a control inputgenerated by combining at least the above input components, it ispossible to reduce overshooting of the temperature of the heater withrespect to the target temperature, and to allow the temperature of theheater to smoothly follow the target temperature. As a result, thetemperature of the heater of the exhaust gas sensor can stably becontrolled at the target temperature and the temperature of the activeelement can stably be controlled at a temperature corresponding to thetarget temperature for the heater.

According to the eighth aspect of the present invention, the heatertemperature data representative of the temperature of the heater maydirectly be detected and acquired using a temperature sensor, or may beestimated and acquired based on a suitable parameter or a model, as withthe fourth aspect.

It is preferable to combine two or more of the first through fourthaspects of the present invention with respect to any of the temperaturecontrol apparatus, the temperature control method, and the recordingmedium. If the temperature control apparatus according to the first andsecond aspects are combined, then the temperature control apparatusaccording to the first aspect is characterized by means for sequentiallyacquiring exhaust gas temperature data representing the temperature ofthe exhaust gas, wherein the control input generated by the heatercontrol means includes an input component depending on the temperatureof the exhaust gas represented by the exhaust gas temperature data.Similarly, if the temperature control methods according to the first andsecond aspects are combined, then the temperature control methodaccording to the first aspect is characterized by the step ofsequentially acquiring exhaust gas temperature data representing thetemperature of the exhaust gas, wherein when the control input isgenerated, the control input is generated so as to further include aninput component depending on the temperature of the exhaust gasrepresented by the exhaust gas temperature data. Furthermore, if therecording mediums according to the first and second aspects arecombined, then the recording medium according to the first aspect ischaracterized in that the temperature control program further includes aprogram for enabling the computer to perform a process of sequentiallyacquiring exhaust gas temperature data representing the temperature ofthe exhaust gas, wherein the control input generating program has analgorithm for enabling the computer to generate the control input so asto further include an input component depending on the temperature ofthe exhaust gas represented by the exhaust gas temperature data.

By thus combining the first and second aspects, the advantages of thoseaspects are added together for controlling the temperature of the activeelement more stably at the target temperature.

In the temperature control apparatus according to the first aspect orthe combination thereof with the temperature control apparatus accordingto the second aspect, the control input generated by the heater controlmeans may include an input component depending on the targettemperature, providing an invention based on the combination of thetemperature control apparatus according to the first and third aspectsor an invention based on the combination of the temperature controlapparatus according to the first through third aspects. Similarly, inthe temperature control method according to the first aspect or thecombination thereof with the temperature control method according to thesecond aspect, when the control input is generated, the control inputmay be generated so as to further include an input component dependingon the target temperature, providing an invention based on thecombination of the temperature control methods according to the firstand third aspects or an invention based on the combination of thetemperature control methods according to the first through thirdaspects. Furthermore, in the recording medium according to the firstaspect or the combination thereof with the recording medium according tothe second aspect, the control input generating program may have analgorithm for enabling the computer to generate the control input so asto further include an input component depending on the targettemperature, providing an invention based on the combination of therecording mediums according to the first and third aspects or aninvention based on the combination of the recording mediums according tothe first through third aspects.

If the first and third aspects are thus combined, the advantages ofthose aspects are added together for controlling the temperature of theactive element more stably at the target temperature. Particularly, ifthe first through third aspects are combined, the advantages of thoseaspects are added together for appropriately increasing the stability ofthe temperature of the active element with respect to the targettemperature.

Moreover, in the temperature control apparatus according to the firstaspect or the combination thereof with the temperature control apparatusaccording to one or more of the second and third aspects, the controlinput generated by the heater control means may include an inputcomponent depending on the temperature of the active element representedby the element temperature data, providing an invention based on thecombination of the temperature control apparatus according to the firstand fourth aspects or an invention based on the combination thereof withthe temperature control apparatus according to one or more of the secondand third aspects. Similarly, in the temperature control methodaccording to the first aspect or the combination thereof with thetemperature control method according to one or more of the second andthird aspects, when the control input is generated, the control inputmay be generated so as to further include an input component dependingon the temperature of the active element represented by the elementtemperature data, providing an invention based on the combination of thetemperature control methods according to the first and fourth aspects oran invention based on the combination thereof with the temperaturecontrol method according to one or more of the second and third aspects.Furthermore, in the recording medium according to the first aspect orthe combination thereof with the recording medium according to one ormore of the second and third aspects, the control input generatingprogram may have an algorithm for enabling the computer to generate thecontrol input so as to further include an input component depending onthe temperature of the active element represented by the elementtemperature data, providing an invention based on the combination of therecording mediums according to the first and fourth aspects or aninvention based on the combination thereof with the recording mediumaccording to one or more of the second and third aspects.

By thus combining two or more of the first through fourth aspects, theadvantages of those aspects are added together for increasing thestability of the temperature of the active element with respect to thetarget temperature.

In the temperature control apparatus according to the second aspect, thecontrol input generated by the heater control means may include an inputcomponent depending on the target temperature, providing an inventionbased on the combination of the temperature control apparatus accordingto the second and third aspects. Similarly, in the temperature controlmethod according to the second aspect, when the control input isgenerated, the control input may be generated so as to further includean input component depending on the target temperature, providing aninvention based on the combination of the temperature control methodsaccording to the second and third aspects. Furthermore, in the recordingmedium according to the second aspect, the control input generatingprogram may have an algorithm for enabling the computer to generate thecontrol input so as to further include an input component depending onthe target temperature, providing an invention based on the combinationof the recording mediums according to the second and third aspects.

With the above arrangement, the advantages of the second and thirdaspects are added together for controlling the temperature of the activeelement more stably at the target temperature.

In the temperature control apparatus according to the second aspect orthe combination thereof with the temperature control apparatus accordingto the third aspect, the control input generated by the heater controlmeans may include an input component depending on the temperature of theactive element represented by the element temperature data, providing aninvention based on the combination of the temperature control apparatusaccording to the second and fourth aspects or an invention based on thecombination of the temperature control apparatus according to the secondthrough fourth aspects. Similarly, in the temperature control methodaccording to the second aspect or the combination thereof with thetemperature control method according to the third aspect, when thecontrol input is generated, the control input may be generated so as tofurther include an input component depending on the temperature of theactive element represented by the element temperature data, providing aninvention based on the combination of the temperature control methodsaccording to the second and fourth aspects or an invention based on thecombination of the temperature control methods according to the secondthrough fourth aspects. Furthermore, in the recording medium accordingto the second aspect or the combination thereof with the recordingmedium according to the third aspect, the control input generatingprogram may have an algorithm for enabling the computer to generate thecontrol input so as to further include an input component depending onthe temperature of the active element represented by the elementtemperature data, providing an invention based on the combination of therecording mediums according to the second and fourth aspects or aninvention based on the combination of the recording mediums according tothe second through fourth aspects.

By thus combining those aspects, the advantages of the aspects are addedtogether for increasing the stability of the temperature of the activeelement with respect to the target temperature.

In the temperature control apparatus according to the third aspect, thecontrol input generated by the heater control means may include an inputcomponent depending on the temperature of the active element representedby the element temperature data, providing an invention based on thecombination of the temperature control apparatus according to the thirdand fourth aspects. Similarly, in the temperature control methodaccording to the third aspect, when the control input is generated, thecontrol input may be generated so as to further include an inputcomponent depending on the temperature of the active element representedby the element temperature data, providing an invention based on thecombination of the temperature control methods according to the thirdand fourth aspects. Furthermore, in the recording medium according tothe third aspect, the control input generating program may have analgorithm for enabling the computer to generate the control input so asto further include an input component depending on the temperature ofthe active element represented by the element temperature data,providing an invention based on the combination of the recording mediumsaccording to the third and fourth aspects.

With the above combination, the advantages of the third and fourthaspects are added together for increasing the stability of thetemperature of the active element with respect to the targettemperature.

According to the present invention, it is preferable to combine all thefirst through third aspects or all the first through fourth aspects. Ifall the first through third aspects are combined, then the inputcomponent depending on the difference between the temperature of theactive element and the target temperature should preferably comprise acomponent proportional to an accumulative sum (integrated value) ofvalues of the difference at respective times and a componentproportional to the difference, for example. If all the first throughfourth aspects are combined, then the input component depending on thedifference between the temperature of the active element and the targettemperature should preferably comprise a component proportional to anaccumulative sum (integrated value) of values of the difference atrespective times.

The combination of the inventions according to the first through fourthaspects is also applicable to the inventions according to the fifththrough eighth aspects, and two or more of the fifth through eighthaspects should preferably be combined. Specifically, if the temperaturecontrol apparatus according to the fifth and sixth aspects are combined,then the temperature control apparatus according to the fifth aspect ischaracterized by means for sequentially acquiring exhaust gastemperature data representing the temperature of the exhaust gas,wherein the control input generated by the heater control means includesan input component depending on the temperature of the exhaust gasrepresented by the exhaust gas temperature data. Similarly, if thetemperature control methods according to the fifth and sixth aspects arecombined, then the temperature control method according to the fifthaspect is characterized by the step of sequentially acquiring exhaustgas temperature data representing the temperature of the exhaust gas,wherein when the control input is generated, the control input isgenerated so as to further include an input component depending on thetemperature of the exhaust gas represented by the exhaust gastemperature data. Furthermore, if the recording mediums according to thefifth and sixth aspects are combined, then the recording mediumaccording to the fifth aspect is characterized in that the temperaturecontrol program further includes a program for enabling the computer toperform a process of sequentially acquiring exhaust gas temperature datarepresenting the temperature of the exhaust gas, wherein the controlinput generating program has an algorithm for enabling the computer togenerate the control input so as to further include an input componentdepending on the temperature of the exhaust gas represented by theexhaust gas temperature data.

By thus combining the fifth and sixth aspects, the advantages of thoseaspects are added together for controlling the temperature of the activeelement more stably at the target temperature and for controlling thetemperature of the active element more stably at a temperaturecorresponding to the target temperature for the heater.

In the temperature control apparatus according to the fifth aspect orthe combination thereof with the temperature control apparatus accordingto the sixth aspect, the control input generated by the heater controlmeans may include an input component depending on the target temperature(a target value for the temperature of the heater), providing aninvention based on the combination of the temperature control apparatusaccording to the fifth and seventh aspects or an invention based on thecombination of the temperature control apparatus according to the fifththrough seventh aspects. Similarly, in the temperature control methodaccording to the fifth aspect or the combination thereof with thetemperature control method according to the sixth aspect, when thecontrol input is generated, the control input may be generated so as tofurther include an input component depending on the target temperature,providing an invention based on the combination of the temperaturecontrol methods according to the fifth and seventh aspects or aninvention based on the combination of the temperature control methodsaccording to the fifth through seventh aspects. Furthermore, in therecording medium according to the fifth aspect or the combinationthereof with the recording medium according to the sixth aspect, thecontrol input generating program may have an algorithm for enabling thecomputer to generate the control input so as to further include an inputcomponent depending on the target temperature, providing an inventionbased on the combination of the recording mediums according to the fifthand sixth aspects or an invention based on the combination of therecording mediums according to the fifth through seventh aspects.

If the fifth and seventh aspects are thus combined, the advantages ofthose aspects are added together for controlling the temperature of theheater more stably at the target temperature and for controlling thetemperature of the active element more stably at a temperaturecorresponding to the target temperature for the heater. Particularly, ifthe fifth through seventh aspects are combined, the advantages of thefifth through seventh aspects are added together for appropriatelyincreasing the stability of the temperature of the heater and thetemperature of the active element.

Moreover, in the temperature control apparatus according to the fifthaspect or the combination thereof with the temperature control apparatusaccording to one or more of the sixth and seventh aspects, the controlinput generated by the heater control means may include an inputcomponent depending on the temperature of the heater represented by theheater temperature data, providing an invention based on the combinationof the temperature control apparatus according to the fifth and eighthaspects or an invention based on the combination thereof with thetemperature control apparatus according to one or more of the sixth andseventh aspects. Similarly, in the temperature control method accordingto the fifth aspect or the combination thereof with the temperaturecontrol method according to one or more of the sixth and seventhaspects, when the control input is generated, the control input may begenerated so as to further include an input component depending on thetemperature of the heater represented by the heater temperature data,providing an invention based on the combination of the temperaturecontrol methods according to the fifth and eighth aspects or aninvention based on the combination thereof with the temperature controlmethod according to one or more of the sixth and seventh aspects.Furthermore, in the recording medium according to the fifth aspect orthe combination thereof with the recording medium according to one ormore of the sixth and seventh aspects, the control input generatingprogram may have an algorithm for enabling the computer to generate thecontrol input so as to further include an input component depending onthe temperature of the heater represented by the heater temperaturedata, providing an invention based on the combination of the recordingmediums according to the fifth and eighth aspects or an invention basedon the combination thereof with the recording medium according to one ormore of the sixth and seventh aspects.

By thus combining those aspects, the advantages of the aspects are addedtogether for increasing the stability of the temperature of the heaterwith respect to the target temperature and for controlling thetemperature of the active element more stably at a temperaturecorresponding to the target temperature for the heater.

In the temperature control apparatus according to the sixth aspect, thecontrol input generated by the heater control means may include an inputcomponent depending on the target temperature (a target value for thetemperature of the heater), providing an invention based on thecombination of the temperature control apparatus according to the sixthand seventh aspects. Similarly, in the temperature control methodaccording to the sixth aspect, when the control input is generated, thecontrol input may be generated so as to further include an inputcomponent depending on the target temperature, providing an inventionbased on the combination of the temperature control methods according tothe sixth and seventh aspects. Furthermore, in the recording mediumaccording to the sixth aspect, the control input generating program mayhave an algorithm for enabling the computer to generate the controlinput so as to further include an input component depending on thetarget temperature, providing an invention based on the combination ofthe recording mediums according to the sixth and seventh aspects.

With the above arrangement, the advantages of the sixth and seventhaspects are added together for controlling the temperature of the heatermore stably at the target temperature and for controlling thetemperature of the active element more stably at a temperaturecorresponding to the target temperature for the heater.

Moreover, in the temperature control apparatus according to the sixthaspect or the combination thereof with the temperature control apparatusaccording to the seventh aspect, the control input generated by theheater control means may include an input component depending on thetemperature of the heater represented by the heater temperature data,providing an invention based on the combination of the temperaturecontrol apparatus according to the sixth and eighth aspects or aninvention based on the combination thereof with the temperature controlapparatus according to the sixth through eighth aspects. Similarly, inthe temperature control method according to the sixth aspect or thecombination thereof with the temperature control method according to theseventh aspect, when the control input is generated, the control inputmay be generated so as to further include an input component dependingon the temperature of the heater represented by the heater temperaturedata, providing an invention based on the combination of the temperaturecontrol methods according to the sixth and eighth aspects or aninvention based on the combination thereof with the temperature controlmethod according to the sixth through eighth aspects. Furthermore, inthe recording medium according to the sixth aspect or the combinationthereof with the recording medium according to the seventh aspect, thecontrol input generating program may have an algorithm for enabling thecomputer to generate the control input so as to further include an inputcomponent depending on the temperature of the heater represented by theheater temperature data, providing an invention based on the combinationof the recording mediums according to the sixth and eighth aspects or aninvention based on the combination thereof with the recording mediumaccording to the sixth through eighth aspects.

By thus combining those aspects, the advantages of the aspects are addedtogether for increasing the stability of the temperature of the heaterwith respect to the target temperature and for controlling thetemperature of the active element more stably at a temperaturecorresponding to the target temperature for the heater.

Moreover, in the temperature control apparatus according to the seventhaspect, the control input generated by the heater control means mayinclude an input component depending on the temperature of the heaterrepresented by the heater temperature data, providing an invention basedon the combination of the temperature control apparatus according to theseventh and eighth aspects. Similarly, in the temperature control methodaccording to the seventh aspect, when the control input is generated,the control input may be generated so as to further include an inputcomponent depending on the temperature of the heater represented by theheater temperature data, providing an invention based on the combinationof the temperature control methods according to the seventh and eighthaspects. Furthermore, in the recording medium according to the seventhaspect, the control input generating program may have an algorithm forenabling the computer to generate the control input so as to furtherinclude an input component depending on the temperature of the heaterrepresented by the heater temperature data, providing an invention basedon the combination of the recording mediums according to the seventh andeighth aspects.

With this combination, the advantages of the seventh and eighth aspectsare added together for increasing the stability of the temperature ofthe heater with respect to the target temperature and for controllingthe temperature of the active element more stably at a temperaturecorresponding to the target temperature for the heater.

It is preferable to combine all the fifth through seventh aspects or allthe fifth through eighth aspects in particular. If all the fifth throughseventh aspects are combined, then the input component depending on thedifference between the temperature of the heater and the targettemperature should preferably comprise a component proportional to anaccumulative sum (integrated value) of values of the difference atrespective times and a component proportional to the difference, forexample. If all the fifth through eighth aspects are combined, then theinput component depending on the difference between the temperature ofthe heater and the target temperature should preferably comprise acomponent proportional to an accumulative sum (integrated value) ofvalues of the difference at respective times.

In the temperature control apparatus according to the second aspectwherein the control input includes an input component depending on thetemperature of the exhaust gas or in the invention based on thecombination of the temperature control apparatus according to one ormore of the first, third, and fourth aspects and the temperature controlapparatus according to the second aspect, the input component dependingon the temperature of the exhaust gas, in the control input sequentiallygenerated by the heater control means, comprises an input componentdepending on time-series data of the temperature of the exhaust gasincluding a present value of the temperature of the exhaust gas and afuture value of the temperature of the exhaust gas after a firstpredetermined time, wherein the heater control means preferablygenerates the control input including the input component according to apredictive control algorithm. Similarly, in the temperature controlmethod according to the second aspect wherein the control input includesan input component depending on the temperature of the exhaust gas or inthe invention based on the combination of the temperature control methodaccording to one or more of the first, third, and fourth aspects and thetemperature control method according to the second aspect, the inputcomponent depending on the temperature of the exhaust gas, included inthe control input, comprises an input component depending on time-seriesdata of the temperature of the exhaust gas including a present value ofthe temperature of the exhaust gas and a future value of the temperatureof the exhaust gas after a first predetermined time, wherein the controlinput including the input component is preferably generated according toa predictive control algorithm. Furthermore, in the recording mediumaccording to the second aspect wherein the control input includes aninput component depending on the temperature of the exhaust gas or inthe invention based on the combination of the recording medium accordingto one or more of the first, third, and fourth aspects and the recordingmedium according to the second aspect, the input component depending onthe temperature of the exhaust gas, included in the control input,comprises an input component depending on time-series data of thetemperature of the exhaust gas including a present value of thetemperature of the exhaust gas and a future value of the temperature ofthe exhaust gas after a first predetermined time, wherein the controlinput generating program for enabling the computer to generate thecontrol input including the input component preferably has a predictivecontrol algorithm.

The foregoing also applies to the invention according to the sixthaspect or the invention based on the combination of one or more of thefifth, seventh, and eighth aspects and the sixth aspect, with respect toany of the temperature control apparatus, the temperature controlmethod, and the recording medium according to the present invention forgenerating a control input to equalize the temperature of the heater tothe target temperature.

With the above arrangement, the input component included in the controlinput which depends on the temperature of the exhaust gas is an inputcomponent depending on not only the present value of the temperature ofthe exhaust gas, but also the time-series data of the temperature of theexhaust gas including at least the future value after the firstpredetermined time. Therefore, the invention according to the secondaspect or an invention including the same is capable of minimizing achange of the temperature of the active element with respect to a changeof the temperature of the exhaust gas. Therefore, the stability withwhich to control the temperature of the active element at the targettemperature can effectively be increased irrespective of variations ofthe temperature of the exhaust gas. Similarly, the invention accordingto the sixth aspect or an invention including the same is capable ofminimizing a change of the temperature of the heater with respect to achange of the temperature of the exhaust gas. Therefore, the stabilitywith which to control the temperature of the heater at the targettemperature can effectively be increased irrespective of variations ofthe temperature of the exhaust gas, and the temperature of the activeelement can more stably be controlled at a temperature corresponding tothe target temperature for the heater. The time-series data may includenot only the present value of the temperature of the exhaust gas and thefuture value thereof after the first predetermined time, but also aplurality of future values of the temperature of the exhaust gas frompresent to a time after the first predetermined time.

If the control input is generated according to the predictive controlalgorithm taking into account the future value of the temperature of theexhaust gas, then the future value of the temperature of the exhaust gaswhich is required to generate the control input may be estimatedaccording to a suitable algorithm. However, in general, the temperatureof the exhaust gas does not change appreciably abruptly. In thetemperature control apparatus according to the second aspect or thesixth aspect, the heater control means may generate the control input bydetermining the future value of the temperature of the exhaust gas untilafter the first predetermined time as being equal to the present valueof the temperature of the exhaust gas. Similarly, in the temperaturecontrol method according to the second aspect or the sixth aspect, thepredictive control algorithm may comprise an algorithm for generatingthe control input by determining the future value of the temperature ofthe exhaust gas until after the first predetermined time as being equalto the present value of the temperature of the exhaust gas. Furthermore,in the recording medium according to the second aspect or the sixthaspect, the algorithm of the control input generating program may enablethe computer to generate the control input by determining the futurevalue of the temperature of the exhaust gas until after the firstpredetermined time as being equal to the present value of thetemperature of the exhaust gas.

With the above arrangement, since the present value of the temperatureof the exhaust gas may sequentially be grasped, the algorithm forcontrolling the temperature of the exhaust gas sensor, including aprocess of grasping the temperature of the exhaust gas (acquiring thedata representing the temperature of the exhaust gas), can easily beconstructed.

In the temperature control apparatus according to the third aspectwherein the control input includes an input component depending on thetarget temperature for the active element or in the invention based onthe combination of the temperature control apparatus according to one ormore of the first, second, and fourth aspects and the temperaturecontrol apparatus according to the third aspect, the input componentdepending on the target temperature which is sequentially generated bythe heater control means comprises an input component depending ontime-series data of the target temperature including a present value ofthe target temperature and a future value of the target temperatureafter a second predetermined time, wherein the heater control meanspreferably generates the control input including the input componentaccording to a predictive control algorithm. Similarly, in thetemperature control method according to the third aspect or in theinvention based on the combination of the temperature control methodaccording to one or more of the first, second, and fourth aspects andthe temperature control method according to the third aspect, the inputcomponent depending on the target temperature, included in the controlinput, comprises an input component depending on time-series data of thetarget temperature including a present value of the target temperatureand a future value of the target temperature after a secondpredetermined time, wherein the control input including the inputcomponent is preferably generated according to a predictive controlalgorithm. Furthermore, in the recording medium according to the thirdaspect or in the invention based on the combination of the recordingmedium according to one or more of the first, second, and fourth aspectsand the recording medium according to the third aspect, the inputcomponent depending on the target temperature, included in the controlinput, comprises an input component depending on time-series data of thetarget temperature including a present value of the target temperatureand a future value of the target temperature after a secondpredetermined time, wherein the control input generating program forenabling the computer to generate the control input including the inputcomponent preferably has a predictive control algorithm.

The foregoing also applies to the invention according to the seventhaspect or the invention based on the combination of one or more of thefifth, sixth, and eighth aspects and the seventh aspect, with respect toany of the temperature control apparatus, the temperature controlmethod, and the recording medium according to the present invention forgenerating a control input to equalize the temperature of the heater tothe target temperature.

With the above arrangement, the input component included in the controlinput which depends on the target temperature is an input componentdepending on not only the present value of the target temperature, butalso the time-series data of the target temperature including at leastthe future value after the second predetermined time. Therefore, theinvention according to the third aspect or an invention including thesame is capable of preventing the temperature of the active element fromovershooting the target temperature therefor when the target temperaturefor the active element is changed. The rate at which the temperature ofthe active element converges to the target temperature can be increased.As a result, the temperature of the active element can quickly andsmoothly follow the target temperature. In particular, immediately afterthe internal combustion engine has started to operate, the temperatureof the active element can quickly be converted to and stabilized at thetarget temperature. Therefore, immediately after the internal combustionengine has started to operate, the output characteristics of the activeelement can quickly be stabilized at desired characteristics. As theability of the temperature of the active element to follow a change ofthe target temperature for the active element is increased, the targettemperature for the active element can variably be set to a value thatis suitable for the operating state of the internal combustion engine.

Similarly, the invention according to the seventh aspect or an inventionincluding the same is capable of preventing the temperature of theheater from overshooting the target temperature therefor when the targettemperature for the heater is changed. The rate at which the temperatureof the heater converges to the target temperature can be increased. As aresult, the temperature of the heater can quickly and smoothly followthe target temperature. In particular, immediately after the internalcombustion engine has started to operate, the temperature of the heatercan quickly be converted to and stabilized at the target temperature,and the temperature of the active element can quickly be converted andstabilized at a temperature corresponding to the target temperature forthe heater. Therefore, immediately after the internal combustion enginehas started to operate, the output characteristics of the active elementcan quickly be stabilized at desired characteristics. As the ability ofthe temperature of the heater to follow a change of the targettemperature for the heater is increased, the target temperature for theheater and a desired temperature of the active element can variably beset to a value that is suitable for the operating state of the internalcombustion engine.

The time-series data of the target temperature may include not only thepresent value of the target temperature and the future value thereofafter the second predetermined time, but also a plurality of futurevalues of the target temperature from present to a time after the secondpredetermined time.

In the temperature control apparatus according to the first throughfourth aspects of the present invention, the heater control meanspreferably generates the control input according to an optimum controlalgorithm. Similarly, in temperature control methods according to thefirst through fourth aspects, the control input is preferably generatedaccording to an optimum control algorithm. In the recording mediumaccording to the first through fourth aspects, the algorithm of thecontrol input generating program preferably comprises an optimum controlalgorithm.

With the above arrangement, according to the first through fourthaspects, it is possible to generate the control input for minimizingvariations of the temperature of the active element and the controlinput while holding in balance the ability of the temperature of theactive element to converge to the target temperature and variations ofthe control input (variations of the heat generating energy supplied tothe heater). As a result, the temperature of the active element can morestably be controlled at the target temperature (desired temperature),and at the same time, the amount of the heat generating energy suppliedto the heater can be held to a required limit.

When the control input is generated according to the optimum controlalgorithm in the first through fourth aspects of the present invention,it is preferable to determine in advance a model to be controlled whichhas, as state quantities to be controlled, the difference between thetemperature of the active element and the target temperature, a changeper given time of the difference (a rate of change of the difference),and a change per given time of the temperature of the heater (a rate ofchange of the temperature of the heater), for example, and which alsohas, as an input quantity, at least a change per given time of thecontrol input (a rate of change of the control input), and to constructthe optimum control algorithm based on the model to be controlled.Alternatively, it is also preferable to determine in advance a model tobe controlled which has, as state quantities to be controlled, thedifference between the temperature of the active element and the targettemperature, a change per given time of the temperature of the activeelement (a rate of change of the temperature of the active element), anda change per given time of the temperature of the heater (a rate ofchange of the temperature of the heater), for example, and which alsohas, as an input quantity, at least a change per given time of thecontrol input (a rate of change of the control input), and to constructthe optimum control algorithm based on the model to be controlled. Atany rate, there is determined a control input capable of minimizing anevaluating function (which is represented as an integrated value(accumulated sum) of weighted sum of the squares of the above statequantities and the square of the change per given time of the controlinput) of the model to be controlled. With this arrangement, since thecontrol input can be generated in a manner to not only eliminate thedifference between the temperature of the active element and the targettemperature, but also suppress variations of the state quantities of themodel to be controlled and the control input while keeping them inbalance, the stability of the temperature of the active element withrespect to the target temperature can be increased.

If the model to be controlled, which serves as a basis for the optimumcontrol algorithm in the first through fourth aspects has, as statequantities to be controlled, the difference between the temperature ofthe active element and the target temperature, the change per given timeof the difference, and the change per given time of the temperature ofthe heater, it is possible to generate the control input which includesthe input component depending on the difference between the temperatureof the active element and the target temperature and the input componentdepending on the temperature of the heater. If the model to becontrolled has, as state quantities to be controlled, the differencebetween the temperature of the active element and the targettemperature, the change per given time of the temperature of the activeelement, and the change per given time of the temperature of the heater,it is possible to generate the control input which includes the inputcomponent depending on the difference between the temperature of theactive element and the target temperature, the input component dependingon the temperature of the heater, and the input component depending onthe temperature of the active element. By including, as input quantitiesfor the model to be controlled, a change per given time of thetemperature of the exhaust gas and a change per given time of the targettemperature for the temperature of the active element, in addition tothe change per given time of the control input, it is possible togenerate the control input including the input component depending onthe temperature of the exhaust gas and the input component depending onthe target temperature for the active element.

In the temperature control apparatus according to the fifth througheighth aspects of the present invention, the heater control meanspreferably generates the control input according to an optimum controlalgorithm. Similarly, in temperature control methods according to thefifth through eighth aspects, the control input is preferably generatedaccording to an optimum control algorithm. In the recording mediumaccording to the fifth through eighth aspects, the algorithm of thecontrol input generating program preferably comprises an optimum controlalgorithm.

With the above arrangement, it is possible to generate the control inputfor minimizing variations of the temperature of the heater and thecontrol input while holding in balance the ability of the temperature ofthe heater to converge to the target temperature and variations of thecontrol input. As a result, the temperature of the heater can morestably be controlled at the target temperature and the temperature ofthe active element can more stably be controlled at a temperature(desired temperature) corresponding to the target temperature for theheater. At the same time, the amount of the heat generating energysupplied to the heater can be held to a required limit.

When the control input is generated according to the optimum controlalgorithm in the fifth through eighth aspects of the present invention,it is preferable to determine in advance a model to be controlled whichhas, as state quantities to be controlled, the difference between thetemperature of the heater and the target temperature, a change per giventime of the difference (a rate of change of the difference), and achange per given time of the temperature of the active element (a rateof change of the temperature of the active element), for example, andwhich also has, as an input quantity, at least a change per given timeof the control input (a rate of change of the control input), and toconstruct the optimum control algorithm based on the model to becontrolled. Alternatively, it is also preferable to determine in advancea model to be controlled which has, as state quantities to becontrolled, the difference between the temperature of the heater and thetarget temperature, a change per given time of the temperature of theactive element (a rate of change of the temperature of the activeelement), and a change per given time of the temperature of the heater(a rate of change of the temperature of the heater), for example, andwhich also has, as an input quantity, at least a change per given timeof the control input (a rate of change of the control input), and toconstruct the optimum control algorithm based on the model to becontrolled. At any rate, there is determined a control input capable ofminimizing an evaluating function of the model to be controlled. Withthis arrangement, since the control input can be generated in a mannerto not only eliminate the difference between the temperature of theheater and the target temperature, but also suppress variations of thedifference, the temperature of the active element, and the control inputwhile keeping them in balance, the stability of the temperature of theheater with respect to the target temperature and the stability of thetemperature of the active element can be increased.

If the model to be controlled, which serves as a basis for the optimumcontrol algorithm in the fifth through eighth aspects has, as statequantities to be controlled, the difference between the temperature ofthe heater and the target temperature, the change per given time of thedifference, and the change per given time of the temperature of theactive element, it is possible to generate the control input whichincludes the input component depending on the difference between thetemperature of the heater and the target temperature and the inputcomponent depending on the temperature of the active element. If themodel to be controlled has, as state quantities to be controlled, thedifference between the temperature of the heater and the targettemperature, the change per given time of the temperature of the activeelement, and the change per given time of the temperature of the heater,it is possible to generate the control input which includes the inputcomponent depending on the difference between the temperature of theheater and the target temperature, the input component depending on thetemperature of the heater, and the input component depending on thetemperature of the active element. By including, as input quantities forthe model to be controlled, a change per given time of the temperatureof the exhaust gas and a change per given time of the target temperaturefor the temperature of the heater, in addition to the change per giventime of the control input, it is possible to generate the control inputincluding the input component depending on the temperature of theexhaust gas and the input component depending on the target temperaturefor the heater.

When the optimum control algorithm according to the first through fourthaspects of the present invention and the predictive control algorithmfor the temperature of the exhaust gas or the target temperature arecombined, the control input is generated according to an optimumpredictive control algorithm. When the control input is generatedaccording to an optimum predictive control algorithm, the temperature ofthe active element can be controlled at the target temperature with highstability while keeping in balance and suppressing variations of thestate quantities of the model to be controlled, which serves as a basisfor the optimum control algorithm, and the input quantity. The optimumpredictive control algorithm can be constructed based on a model to becontrolled which includes a change per given time of the temperature ofthe exhaust gas and/or a change per given time of the target temperaturefor the active element, in addition to the change per given time of thecontrol input, as the input quantities of the model to be controlledwhich has been given by way of example with respect to the first throughfourth aspects, for example.

The foregoing also applies to a combination of the optimum controlalgorithm according to the fifth through eighth aspects of the presentinvention and the predictive control algorithm for the temperature ofthe exhaust gas or the target temperature. In this case, the controlinput is generated according to an optimum predictive control algorithm.When the control input is generated according to an optimum predictivecontrol algorithm, the temperature of the heater can be controlled atthe target temperature with high stability while keeping in balance andsuppressing variations of the state quantities of the model to becontrolled, which serves as a basis for the optimum control algorithm,and the input quantity. The optimum predictive control algorithm in thiscase can be constructed based on a model to be controlled which includesa change per given time of the temperature of the exhaust gas and/or achange per given time of the target temperature, in addition to thechange per given time of the control input, as the input quantities ofthe model to be controlled which has been given by way of example withrespect to the fifth through eighth aspects, for example.

In the temperature control apparatus, the temperature control method,and the recording medium according to the present invention as describedabove in any of the aspects, the target temperature (a targettemperature for the active element or the heater) in a periodimmediately after the internal combustion engine has started to operateuntil a third predetermined time elapses is preferably set to atemperature which is lower than the target temperature after elapse ofthe period immediately after the internal combustion engine has startedto operate.

Specifically, if the target temperature for the active element or theheater is high immediately after the internal combustion engine hasstarted to operate, then when water in the exhaust passage is applied tothe active element of the exhaust gas sensor, the active element tendsto be damaged due to thermal stresses or the like which are caused byrapid heating of the active element. According to the present invention,the target temperature for the active element or the heater immediatelyafter the internal combustion engine has started to operate is set to atemperature which is lower than the target temperature after elapse ofthe period immediately after the internal combustion engine has startedto operate. In this manner, the active element of the exhaust gas sensoris prevented from being damaged by thermal stresses or the like.

In the temperature control apparatus according to the present inventionas described above in any of the aspects, the heater comprises anelectric heater for generating heat when energized by a batteryaccording to a pulse width control (PWM control) process, if the controlinput generated by the heater control means comprises a duty cycle (theratio of a pulse duration to one period of a pulsed signal used in thepulse with control process) in the pulse width control process, thetemperature control apparatus is preferably further characterized bymeans for correcting the duty cycle depending on the voltage of thebattery. Similarly, in the temperature control method according to thepresent invention in any of the aspects, the heater comprises anelectric heater for generating heat when energized by a batteryaccording to a pulse width control (PWM control) process, and thecontrol input which is generated comprises a duty cycle in the pulsewidth control process, the temperature control method is preferablyfurther characterized by the step of correcting the duty cycle dependingon the voltage of the battery. Furthermore, in the recording mediumaccording to the present invention in any of the aspects, the heatercomprises an electric heater for generating heat when energized by abattery according to a pulse width control (PWM control) process, andthe control input which is generated by the computer according to thecontrol input program comprises a duty cycle in the pulse width controlprocess, wherein the temperature control program preferably furtherincludes a program for enabling the computer to perform a process ofcorrecting the duty cycle generated by the control input generatingprogram depending on the voltage of the battery.

Specifically, if the voltage of the battery is substantially constant,then the duty cycle defines an amount of electric power as an amount ofheat generating energy supplied to the heater. If the voltage of thebattery is varied by an alternator or the like, then the amount ofelectric power supplied to the heater is affected by not only the dutycycle but also the voltage of the battery. Therefore, by correcting theduty cycle as the control input depending on the voltage of the battery,variations of the voltage of the battery in controlling the temperatureof the active element or the heater at the target temperature can becompensated for. As a result, the temperature of the active element orthe temperature of the heater can stably be controlled at the targettemperature without being affected by variations of the voltage of thebattery, and the stability of the temperature of the active element canbe increased.

According to the present invention, the exhaust gas sensor may comprisean O₂ sensor disposed downstream of a catalytic converter for purifyingthe exhaust gas, for example. If the air-fuel ratio of the exhaust gasis controlled to keep the output voltage of the O₂ sensor at apredetermined level in order for the catalytic converter to perform itsdesired exhaust gas purifying capability, the temperature of the activeelement of the O₂ sensor should preferably basically be controlled at atemperature equal to or higher than 750° C. (e.g., 800° C.) for betterair-fuel ratio control. In this case, when the heater is to becontrolled with a target temperature determined for the active element,the target temperature may be set to a temperature equal to or higherthan 750° C. (e.g., 800° C.). When the heater is to be controlled with atarget temperature determined for the heater, the target temperature maybe set to a temperature equal to or higher than 850° C. (e.g., 900° C.).

Furthermore, immediately after the internal combustion engine hasstarted to operate (in a period in which a third predetermined time(e.g., 15 seconds) elapses after the internal combustion engine hasstarted to operate), the temperature of the active element shouldpreferably be controlled at a temperature (e.g., 600° C.) lower than theabove temperature in order to prevent the active element of the O₂sensor from being damaged. In this case, when the heater is to becontrolled with a target temperature determined for the active element,the target temperature for the active element immediately after theinternal combustion engine has started to operate may be set to 600° C.,for example. When the heater is to be controlled with a targettemperature determined for the heater, the target temperature for theheater immediately after the internal combustion engine has started tooperate may be set to 700° C., for example.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a block diagram of an apparatus according to a firstembodiment of the present invention;

FIG. 2 is a fragmentary cross-sectional view showing a structure of anO₂ sensor (exhaust gas sensor) in the apparatus shown in FIG. 1;

FIG. 3 is a graph illustrative of the output characteristics of the O₂sensor shown in FIG. 2;

FIG. 4 is a block diagram showing a functional arrangement of a sensortemperature control means in the apparatus shown in FIG. 1;

FIG. 5 is a cross-sectional view showing a processing operation of anexhaust temperature observer in the sensor temperature control meansshown in FIG. 4;

FIG. 6 is a block diagram showing a functional arrangement of theexhaust temperature observer in the sensor temperature control meansshown in FIG. 4;

FIG. 7 is a block diagram showing a functional arrangement of a heatercontroller in the sensor temperature control means shown in FIG. 5;

FIG. 8 is a flowchart of an overall processing sequence of the sensortemperature control means in the apparatus shown in FIG. 1;

FIGS. 9 through 11 are flowcharts of subroutines of the flowchart shownin FIG. 8;

FIGS. 12 and 13 are graphs showing the results of simulations of thefirst embodiment; and

FIG. 14 is a block diagram showing a functional arrangement of a sensortemperature control means in an apparatus according to a secondembodiment of the present invention.

BEST MODE FOR CARRYING OUT THE INVENTION

A first embodiment of the present invention will be described below withreference to FIGS. 1 through 13. FIG. 1 shows in block form an overallarrangement of the apparatus according to the first embodiment of thepresent invention. In FIG. 1, an engine (an internal combustion engine)1 mounted on an automobile, a hybrid vehicle, or the like combusts amixture of fuel and air and generates an exhaust gas, which isdischarged into the atmosphere through an exhaust passage 3communicating with an exhaust port 2 of the engine 1. The exhaustpassage 3 incorporates therein two catalytic converters 4, 5 disposedsuccessively downstream for purifying the exhaust gas emitted from theengine 1 and flowing through the exhaust passage 3. The exhaust passage3 includes a section upstream of the catalytic converter 4 (between theexhaust port 2 and the catalytic converter 4), a section between thecatalytic converters 4, 5, and a section downstream of the catalyticconverter 5. These sections of the exhaust passage 3 are provided byrespective exhaust pipes 6 a, 6 b, 6 c each in the form of a tubularpassage-defining member.

Each of the catalytic converters 4, 5 contains a catalyst 7 (three-waycatalyst in the present embodiment). The catalyst 7 has apassage-defining honeycomb structure and allows the exhaust gas to flowtherethrough. Though the catalytic converters 4, 5 may be of a unitarystructure with two catalytic beds, each comprising a three-way catalyst,disposed respectively in upstream and downstream regions thereof.

In the present embodiment, the air-fuel ratio in the exhaust gas emittedfrom the engine 1 is controlled in order for the upstream catalyticconverter 4, in particular, to have a good exhaust gas purifyingcapability (the ability of the catalytic converter 4 to purify CO, HC,and NOx). For controlling the air-fuel ratio in the exhaust gas, an O₂sensor 8 is mounted on the exhaust passage 3 between the catalyticconverters 4, 5, i.e., on the exhaust passage defined by the exhaustpipe 6 b, and a wide-range air-fuel ratio sensor 9 is mounted on theexhaust passage 3 upstream of the catalytic converter 4, i.e., on theexhaust passage defined by the exhaust pipe 6 a. When the catalyticconverters 4, 5 take a unitary structure with two catalytic beds, the O₂sensor 8 may be mounted between the upstream catalytic bed and thedownstream catalytic bed.

The O₂ sensor 8 corresponds to an exhaust gas sensor according to thepresent invention. Basic structural details and characteristics of theO₂ sensor 8 will be described below. As shown in FIG. 2, the O₂ sensor 8has an active element 10 (sensitive element) in the form of a hollowbottomed cylinder made primarily of a solid electrolyte permeable tooxygen ions, e.g., stabilized zirconia (ZrO₂+Y₂O₃). The active element10 has outer and inner surfaces coated with porous platinum electrodes11, 12, respectively. The O₂ sensor 8 also has a rod-shaped ceramicheater 13 inserted as an electric heater into the active element 10 forheating the active element 10 for activation and controlling thetemperature of the active element 10. The active element 10 is filledwith air containing oxygen at a constant concentration, i.e., under aconstant partial pressure, in a space around the ceramic heater 13. TheO₂ sensor 8 is placed in a sensor casing 14 mounted on the exhaust pipe6 b such that the tip end of the active element 10 has its outer surfacepositioned in contact with the exhaust gas flowing in the exhaust pipe 6b.

As shown in FIG. 2, the tip end of the active element 10 is covered witha tubular protector 15 which protects the active element 10 against theimpingement of foreign matter thereon. The tip end of the active element10 which is positioned in the exhaust pipe 6 b contacts the exhaust gasthrough a plurality of holes (not shown) defined in the protector 15.

The O₂ sensor 8 thus constructed operates as follows: An electromotiveforce depending on the concentration of oxygen in the exhaust gas isgenerated between the platinum electrodes 11, 12 based on the differencebetween the concentration of oxygen in the exhaust gas which is broughtinto contact with the outer surface of the tip end of the active element10 and the concentration of oxygen in the air in the active element 10.The generated electromotive force is amplified by an amplifier (notshown), and then produced as the output voltage Vout from the O₂ sensor8.

The output voltage Vout of the O₂ sensor 8 has characteristics (outputcharacteristics) with respect to the concentration of oxygen in theexhaust gas or the air-fuel ratio in the exhaust gas which is recognizedfrom the concentration of oxygen, as represented by a solid-line curve“a” (so-called “Z curve”) in FIG. 3. The solid-line curve “a” in FIG. 3represents the output characteristics of the O₂ sensor 8 when thetemperature of the active element 10 is 800° C. The relationship betweenthe temperature of the active element 10 and the output characteristicsof the O₂ sensor 8 will be described later on.

As indicated by the curve “a” in FIG. 3, the output characteristics ofthe O₂ sensor 8 are generally of such a nature that the output voltageVout changes substantially linearly with high sensitivity with respectto the air-fuel ratio of the exhaust gas only when the air-fuel ratiorepresented by the concentration of oxygen in the exhaust gas is presentin a narrow air-fuel ratio range Δ near a stoichiometric air-fuel ratio.In the air-fuel ratio range Δ (hereinafter referred to as“high-sensitivity air-fuel ratio range Δ”), the gradient of a change inthe output voltage Vout with respect to a change in the air-fuel ratio,i.e., the gradient of the curve of the output characteristics of the O₂sensor 8, is large. In an air-fuel ratio range richer than thehigh-sensitivity air-fuel ratio range Δ and an air-fuel ratio rangeleaner than the high-sensitivity air-fuel ratio range Δ, the gradient ofa change in the output voltage Vout with respect to a change in theair-fuel ratio, i.e., the gradient of the curve of the outputcharacteristics of the O₂ sensor 8, is smaller.

The wide-range air-fuel ratio sensor 9, which will not be described indetail below, comprises an air-fuel ratio sensor disclosed in Japaneselaid-open patent publication No. 4-369471 by the applicant of thepresent application, for example. The wide-range air-fuel ratio sensor 9is a sensor for generating an output voltage KACT which changes linearlywith respect to the air-fuel ratio in the exhaust gas in the air-fuelratio wider in range than the O₂ sensor 8. The output voltage Vout ofthe O₂ sensor 8 and the output voltage KACT of the wide-range air-fuelratio sensor 9 will hereinafter be referred to as “output Vout” and“output KACT”, respectively.

The apparatus according to the present embodiment also has a controlunit 16 for controlling the air-fuel ratio in the exhaust gas andcontrolling the temperature of the active element 10 of the O₂ sensor 8.The control unit 16 comprises a microcomputer including a CPU, a RAM,and a ROM (not shown). For carrying out a control process to bedescribed later on, the control unit 16 is supplied with the outputsVout and KACT from the O₂ sensor 8 and the wide-range air-fuel ratiosensor 9, and also with data of detected values representing therotational speed NE of the engine 1, the intake pressure PB(specifically, the absolute pressure in the intake pipe of the engine1), the atmospheric temperature TA, the engine temperature TW(specifically, the temperature of the coolant of the engine 1), etc.,from sensors (not shown) combined with the engine 1. The control unit 16is also supplied with data of detected values representing the voltageVB (hereinafter referred to as “battery voltage VB”) of a battery (notshown) as a power supply of electronic accessories including an ignitionunit (not shown) of the engine 1, the control unit 16, the ceramicheater 13, etc., from a sensor (not shown). The ROM of the control unit16 corresponds to a recording medium according to the present invention.

The control unit 16 has as its functional means an air-fuel ratiocontrol means 17 for controlling the air-fuel ratio in the exhaust gasemitted from the engine 1, and a sensor temperature control means 18 forcontrolling the temperature of the active element 10 of the O₂ sensor 8.

The air-fuel ratio control means 17 controls the air-fuel ratio in theexhaust gas supplied from the engine 1 to the catalytic converter 4 inorder to achieve a good purifying ability (purification rate) of thecatalytic converter 4 to purify CO (carbon monoxide), HC (hydrocarbon),and NOx (nitrogen oxide). When the O₂ sensor 8 of the above outputcharacteristics is disposed downstream of the catalytic converter 4, agood purifying ability of the catalytic converter 4 to purify CO, HC,and NOx can be achieved irrespective of the deteriorated state of thecatalytic converter 4 by controlling the air-fuel ratio in the exhaustgas supplied to the catalytic converter 4, i.e., the air-fuel ratio inthe exhaust gas upstream of the catalytic converter 4, to settle theoutput Vout of the O₂ sensor 8 at a certain predetermined value Vop (seeFIG. 3).

Specifically, the air-fuel ratio control means 17 uses the predeterminedvalue Vop as a target value for the output Vout of the O₂ sensor 8, andcontrols the air-fuel ratio in the exhaust gas supplied from the engine1 to the catalytic converter 4 in order to settle and keep the outputVout of the O₂ sensor 8 at the target value Vop. Such an air-fuel ratiocontrol process is carried out by determining a target air-fuel ratio inthe exhaust gas supplied to the catalytic converter 4 according to afeedback control process in order to converge the output Vout of the O₂sensor 8 to the target value Vop, and adjusting the amount of fuel to besupplied to the engine 1 according to a feedback control process inorder to converge the output KACT (a detected value of the air-fuelratio) of the wide-range air-fuel ratio sensor 9 to the target air-fuelratio. Specific details of the air-fuel ratio control process carried bythe air-fuel ratio control means 17 do not constitute an essentialfeature of the present invention, and will not be described below. Theair-fuel ratio control process carried by the air-fuel ratio controlmeans 17 is carried out as described in paragraphs [0071]-[0362] in thespecification of Japanese laid-open patent publication No. 11-324767 orU.S. Pat. No. 6,188,953, for example.

The output characteristics of the O₂ sensor 8 change depending on thetemperature of the active element 10 thereof. In FIG. 3, the solid-linecurve “a”, a broken-line curve “b”, a dot-and-dash-line curve “c”, and atwo-dot-and-dash-line curve “d” represent the output characteristics ofthe O₂ sensor 8 when the active element 10 has temperatures of 800° C.,750° C., 700° C., and 600° C., respectively. As can be seen from FIG. 3,if the temperature of the active element 10 changes in a temperaturerange lower than 750° C., then the gradient (sensitivity) of a change inthe output Vout of the O₂ sensor 8 in the vicinity of the stoichiometricair-fuel ratio (the high-sensitivity air-fuel ratio range Δ) and thelevel of the output Vout at air-fuel ratios richer than thehigh-sensitivity air-fuel ratio range Δ tend to change. If thetemperature of the active element 10 is 750° C. or higher, then a changein the output characteristics of the O₂ sensor 8 with respect to achange in the temperature of the active element 10 is so small that theoutput characteristics of the O₂ sensor 8 are substantially constant.

Since the output characteristics of the O₂ sensor 8 change depending onthe temperature of the active element 10 as described above, the controlproperties (stability and quick response) of the air-fuel ratio controlmeans 17 are likely to be lowered depending on the temperature of theactive element 10. This is because in controlling the air-fuel ratio inthe exhaust gas in order to keep the output Vout of the O₂ sensor 8 atthe target value Vop, the output characteristics of the O₂ sensor 8 inthe vicinity of the stoichiometric air-fuel ratio, i.e., the outputcharacteristics of the O₂ sensor 8 in the high-sensitivity air-fuelratio range Δ, are liable to greatly affect those control properties.The target value Vop for the output Vout of the O₂ sensor 8 to keep wellthe ability of the catalyst 7 of the catalytic converter 4 to purify theexhaust gas also changes depending on the temperature of the activeelement 10 in a temperature range lower than 750° C. Therefore, it ispreferable to keep the temperature of the active element 10 of the O₂sensor 8 basically at a constant level for the purpose of wellcontrolling the air-air ratio with the air-fuel ratio control means 17,i.e., controlling the output Vout of the O₂ sensor 8 at the target valueVop, and achieving a good purifying ability of the catalytic converter4.

If the temperature of the active element 10 of the O₂ sensor 8 is 750°C. or higher, then the output characteristics of the O₂ sensor 8 aresubstantially constant and stable. According to the inventors'knowledge, if the temperature of the active element 10 is kept at atemperature equal or higher than 750° C., e.g., 800° C., then the targetvalue Vop for the output Vout of the O₂ sensor 8 to keep well theability of the catalyst 7 of the catalytic converter 4 to purify theexhaust gas is present in an area denoted by Y on the curve “a” in FIG.3, i.e., an inflection point Y where the gradient of the curve “a”representing the output characteristics of the O₂ sensor 8 switches froma larger value to a smaller value as the air-fuel ratio becomes richer.At this time, the air-fuel ratio can be controlled to keep the outputVout of the O₂ sensor 8 at the target value Vop. The reason for theabove air-fuel fuel control appears to be that the sensitivity of theout-put Vout of the O₂ sensor 8 to the air-fuel ratio at the inflectionpoint Y is neither excessively high nor small, but is appropriate.

According to the present embodiment, the sensor temperature controlmeans 18 controls the ceramic heater 13 to keep the temperature of theactive element 10 of the O₂ sensor 8 at a desired temperature which isbasically equal to or higher than 750° C., e.g., 800° C. A controlprocess carried out by the sensor temperature control means 18 will bedescribed below.

As shown in FIG. 4, the sensor temperature control means 18 has as itsmajor functions an exhaust temperature observer 19 for sequentiallyestimating an exhaust gas temperature Tgd in the exhaust passage 3 nearthe O₂ sensor 8, i.e., at an intermediate portion of the exhaust pipe 6b, an element temperature observer 20 (temperature estimating means) forestimating the temperature T_(O2) of the active element 10 of the O₂sensor 8 and the temperature Tht of the ceramic heater 13 using theestimated value of the exhaust gas temperature Tgd, a target valuesetting means 21 for setting a target value R (target temperature) forthe temperature of the active element 10, and a heater controller 22(heater control means) for controlling energization of the ceramicheater 13, i.e., controlling the electric energy supplied to the ceramicheater 13, using the estimated values of the temperature T_(O2) of theactive element 10 and the temperature Tht of the ceramic heater 13, thetarget value R, and the estimated value of the exhaust gas temperatureTgd. The exhaust temperature observer 19 is supplied with detected dataof the rotational speed NE of the engine 1, the intake pressure PB, andthe atmospheric temperature TA in order to estimate the exhaust gastemperature Tgd. The heater controller 22 and the element temperatureobserver 20 are supplied with the detected data of the battery voltageVB for performing their processing sequences as described later.

In the present embodiment, the ceramic heater 13 is controlled for itsenergization (PWM control) by giving a pulsed voltage to a heaterenergization circuit (not shown). The amount of electric energy suppliedto the ceramic heater 13 can be controlled by adjusting the duty cycleDUT of the pulsed voltage (the ratio of the pulse duration to one periodof the pulsed voltage). The heater controller 22 sequentially determinesthe duty cycle DUT of the pulsed voltage applied to the heaterenergization circuit as a control input (manipulated variable) forcontrolling the ceramic heater 13, and adjusts the duty cycle DUT tocontrol the amount of electric energy supplied to the ceramic heater 13and hence the amount of heat generated by the ceramic heater 13. Theduty cycle DUT generated by the heater controller 22 is also used in theprocessing sequence of the element temperature observer 20.

According to the present embodiment, the portion of the exhaust passage3 which extends from the exhaust port 2 of the engine 1 to the positionwhere the O₂ sensor 8 is located, i.e., the exhaust passage 3 upstreamof the O₂ sensor 8, is divided into a plurality of (four in the presentembodiment) partial exhaust passageways 3 a, 3 b, 3 c, 3 d along thedirection in which the exhaust passage 3 extends, i.e., the direction inwhich the exhaust gas flows. The exhaust temperature observer 19estimates, in a predetermined cycle time (period), the temperature ofthe exhaust gas at the exhaust port 2 (the inlet of the exhaust passage3) and the temperatures of the exhaust gas in the respective partialexhaust passageways 3 a, 3 b, 3 c, 3 d, or specifically, thetemperatures of the exhaust gas in the downstream ends of the respectivepartial exhaust passageways 3 a, 3 b, 3 c, 3 d, successively in thedownstream direction. Of the partial exhaust passageways 3 a, 3 b, 3 c,3 d, the partial exhaust passageways 3 a, 3 b are two partial exhaustpassageways divided from the exhaust passage 3 upstream of the catalyticconverter 4, i.e., the exhaust passage defined by the exhaust pipe 6 a,the partial exhaust passageway 3 c is a partial exhaust passagewayextending from the inlet to outlet of the catalytic converter 4, i.e.,the exhaust passage defined in the catalyst 7 in the catalytic converter4, and the partial exhaust passageway 3 d is a partial exhaustpassageway extending from the outlet of the catalytic converter 4 to theposition where the O₂ sensor 8 is located, i.e., the exhaust pipe 6 b.The exhaust temperature observer 19 has its algorithm constructed asfollows:

The temperature of the exhaust gas at the exhaust port 2 of the engine 1basically depends on the rotational speed NE and the intake pressure PBof the engine 1 while the engine 1 is operating in a steady state inwhich the rotational speed NE and the intake pressure PB are keptconstant. Therefore, the temperature of the exhaust gas at the exhaustport 2 can basically be estimated from detected values of the rotationalspeed NE and the intake pressure PB, which serve as parametersindicative of the operating state of the engine 1, based on apredetermined map which has been established by way of experimentation,for example. If the operating state (the rotational speed NE and theintake pressure PB) of the engine 1 varies, then the temperature of theexhaust gas at the exhaust port 2 suffers a time lag or delay in theresponse to the exhaust gas temperature determined by the map(hereinafter referred to as “basic exhaust gas temperatureTMAP(NE,PB)”).

According to the present embodiment, the exhaust temperature observer 19determines, in a predetermined cycle time (processing period), the basicexhaust gas temperature TMAP(NE,PB) from the detected values (latestdetected values) of the rotational speed NE and the intake pressure PBof the engine 1 based on the map, and thereafter sequentially estimatesan exhaust gas temperature Texg at the exhaust port 2 as a value whichfollows, with a time lag of first order, the basic exhaust gastemperature TMAP(NE,PB) as expressed by the following equation (1):Texg(k)=(1−Ktex)·Texg(k−1)+Ktex·TMAP(NE,PB)  (1)where k represents the ordinal number of a processing period of theexhaust temperature observer 19, and Ktex a coefficient (lagcoefficient) predetermined by way of experimentation or the like(0<Ktex<1). In the present embodiment, the intake pressure PB of theengine 1 serves as a parameter representative of the amount of intakeair introduced into the engine 1. Therefore, if a flow sensor is usedfor directly detecting the amount of intake air introduced into theengine 1, then the output of the flow sensor, i.e., a detected value ofthe amount of intake air, may be used instead of the detected value ofthe intake pressure PB. In the present embodiment, an initial valueTexg(0) of the estimated value of the exhaust gas temperature Texg isset to the atmospheric temperature TA detected by an atmospherictemperature sensor (not shown) or the engine temperature TW (thetemperature of the coolant of the engine 1) detected by an enginetemperature sensor (not shown) when the engine 1 has started to operate(upon an engine startup), as described later.

Using the estimated value of the exhaust gas temperature Texg at theexhaust port 2, the temperatures of the exhaust gas in the respectivepartial exhaust passageways 3 a, 3 b, 3 c, 3 d are estimated asdescribed below. For illustrative purpose, a general heat transfer thatoccurs when a fluid flows through a circular tube 23 (see FIG. 5) whichextends in the direction of a Z-axis in the atmosphere while exchangingheat with the tube wall of the circular tube 23 will be described below.It is assumed that the fluid temperature Tg and the temperature Tw ofthe tube wall (hereinafter referred to as “circular tube temperatureTw”) are functions Tg(t,z), Tw(t,z) of the time t and the position z inthe direction of the Z-axis, the thermal conductivity of the tube wallof the circular tube 23 is infinite in the radial direction and nil inthe direction of the Z-axis. It is also assumed that the heat transferbetween the fluid and the tube wall of the circular tube 23 and the heattransfer between the tube wall of the circular tube 23 and the externalatmosphere are proportional to their temperature differences accordingto the Newton law of cooling. At this time, the following equations(2-1), (2-2) are satisfied:

$\begin{matrix}{{{{Sg} \cdot \rho}\;{g \cdot {Cg} \cdot \left( {\frac{\partial{Tg}}{\partial t} + {V \cdot \frac{\partial{Tg}}{\partial z}}} \right)}} = {\alpha\;{1 \cdot U \cdot \left( {{Tw} - {Tg}} \right)}}} & \left( {2\text{-}1} \right) \\{{{{Sw} \cdot \rho}\;{w \cdot {Cw} \cdot \frac{\partial{Tw}}{\partial t}}} = {{\alpha\;{1 \cdot U \cdot \left( {{Tg} - {Tw}} \right)}} + {\alpha\;{2 \cdot U \cdot \left( {T_{A} - {Tw}} \right)}}}} & \left( {2\text{-}2} \right)\end{matrix}$where S_(g), ρ_(g), C_(g) represent the density and specific heat of thefluid and the cross-sectional area of the fluid passage, respectively,S_(w), ρ_(w), C_(w) the density, specific heat, and cross-sectional areaof the tube wall of the circular tube 23, respectively, V the speed ofthe fluid flowing through the circular tube 23, T_(A) the atmospherictemperature outside of the circular tube 23, U the inner circumferentiallength of the circular tube 23, α₁ the heat transfer coefficient betweenthe fluid and the tube wall of the circular tube 23, and α₂ the heattransfer coefficient between the tube wall of the circular tube 23 andthe atmosphere. It is assumed that the atmospheric temperature T_(A) iskept constant around the circular tube 23.

The above equations (2-1), (2-2) are modified into the followingequations (3-1), (3-2):

$\begin{matrix}{\frac{\partial{Tg}}{\partial t} = {{{- V} \cdot \frac{\partial{Tg}}{\partial z}} + {a \cdot \left( {{Tw} - {Tg}} \right)}}} & \left( {3\text{-}1} \right) \\{\frac{\partial{Tw}}{\partial t} = {{b \cdot \left( {{Tg} - {Tw}} \right)} + {c \cdot \left( {T_{A} - {Tw}} \right)}}} & \left( {3\text{-}2} \right)\end{matrix}$where a, b, c represent constants, a=α₁·U/(S_(g)·ρ_(g)·C_(g)),b=α₁·U/(S_(w)·ρ_(w)·C_(w)), c=α₂·U/(S_(w)·ρ_(w)·C_(w)).

The first term on the right side of the equation (3-1) is a shiftingflow term representing a time-dependent rate of change of the fluidtemperature Tg (a change in the temperature per unit time) depending onthe temperature gradient in the flowing direction of the fluid and thespeed of the fluid in a position z. The second term on the right side ofthe equation (3-1) is a heat transfer term representing a time-dependentrate of change of the fluid temperature Tg (a change in the temperatureper unit time) depending on the difference between the fluid temperatureTg and the circular tube temperature Tw in the position z, i.e., atime-dependent rate of change of the fluid temperature Tg which iscaused by the heat transfer between the fluid and the tube wall of thecircular tube 23. Therefore, the equation (3-1) indicates that thetime-dependent rate ∂Tg/∂t of change of the fluid temperature Tg in theposition z depends on the temperature change component of the shiftingflow term and the temperature change component of the heat transferterm, i.e., the sum of those temperature change components.

The first term on the right side of the equation (3-2) is a heattransfer term representing a time-dependent rate of change of thecircular tube temperature Tw (a change in the temperature per unit time)depending on the difference between the circular tube temperature Tw andthe fluid temperature Tg in the position z, i.e., a time-dependent rateof change of the circular tube temperature Tw which is caused by theheat transfer between the fluid and the tube wall of the circular tube23 in the position z. The second term on the right side of the equation(3-2) is a heat radiation term representing a time-dependent rate ofchange of the circular tube temperature Tw (a change in the temperatureper unit time) depending on the difference between the circular tubetemperature Tw and the atmospheric temperature TA outside of thecircular tube 23 in the position z, i.e., a time-dependent rate ofchange of the circular tube temperature Tw depending on the heatradiation from the tube wall of the circular tube 23 into the atmospherein the position z. The equation (3-2) indicates that the time-dependentrate ∂Tw/∂t of change of the circular tube temperature Tw in theposition z depends on the temperature change component of the heattransfer term and the temperature change component of the heat radiationterm, i.e., the sum of those temperature change components.

According to the calculus of finite differences, the equations (3-1),(3-2) can be rewritten into the following equations (4-1), (4-2):

$\begin{matrix}{{{Tg}\left( {{t + {\Delta\; t}},z} \right)} = {{{Tg}\left( {t,z} \right)} - {\frac{{V \cdot \Delta}\; t}{\Delta\; z} \cdot \left( {{{Tg}\left( {t,z} \right)} - {{Tg}\left( {t,{z - {\Delta\; z}}} \right)}} \right)} + {{a \cdot \Delta}\;{t \cdot \left( {{{Tw}\left( {t,z} \right)} - {{Tg}\left( {t,z} \right)}} \right)}}}} & \left( {4\text{-}1} \right) \\{{{Tw}\left( {{t + {\Delta\; t}},z} \right)} = {{{Tw}\left( {t,z} \right)} + {{b \cdot \Delta}\;{t \cdot \left( {{{Tg}\left( {t,z} \right)} - {{Tw}\left( {t,z} \right)}} \right)}} + {{c \cdot \Delta}\;{t \cdot \left( {T_{A} - {{Tw}\left( {t,z} \right)}} \right)}}}} & \left( {4\text{-}2} \right)\end{matrix}$

The above equations (4-1), (4-2) indicate that if the fluid temperatureTg(t,z) and the circular tube temperature Tw(t,z) in the position z atthe time t, and the fluid temperature Tg(t,z−Δz) in a position z−Δzwhich precedes the position z (upstream thereof) at the time t areknown, then the fluid temperature Tg(t+Δt,z) and the circular tubetemperature Tw(t+Δt,z) in the position z at a next time t+Δt can bedetermined, and that the fluid temperatures Tg and the circular tubetemperatures Tw in successive positions z+Δz, z+2Δz, . . . can bedetermined by solving the equations (4-1), (4-2) simultaneously insequence for those positions. Specifically, if initial values of thefluid temperature Tg and the circular tube temperature Tw (initialvalues at t=0) are given in the positions z, z+Δz, z+2Δz, . . . and thefluid temperature Tg(t,0) at an origin (e.g., the inlet of the circulartube 23) in the direction of the Z-axis of the circular tube 23 is given(it is assumed that z·Δz=0), then the fluid temperatures Tg and thecircular tube temperatures Tw in successive positions z, z+Δz, z+2Δz, .. . at successive times t, t+Δt, t+2Δt, . . . can be calculated.

The fluid temperature Tg(t,z) in the position z can be calculated bycumulatively adding (integrating), to the initial value Tg(0,z), thetemperature change component depending on the fluid speed V and thetemperature gradient in the position z (the temperature change componentrepresented by the second term of the equation (4-1)) and thetemperature change component depending on the difference between thefluid temperature Tg and the circular tube temperature Tw in theposition z (the temperature change component represented by the thirdterm of the equation (4-1)), at each given time interval. The fluidtemperatures in the other positions z+Δz, z+2Δz, . . . can similarly becalculated. The circular tube temperature Tw(t,z) in the position z canbe calculated by cumulatively adding (integrating), to the initial valueTw(0,z), the temperature change component depending on the differencebetween the fluid temperature Tg and the circular tube temperature Tw inthe position z (the temperature change component represented by thesecond term of the equation (4-2)) and the temperature change componentdepending on the difference between the circular tube temperature Tw andthe atmospheric temperature TA in the position z (the temperature changecomponent represented by the third term of the equation (4-2)), at eachgiven time interval.

In the present embodiment, the exhaust temperature observer 19 uses themodel equations (4-1), (4-2) and determines the temperatures of theexhaust gas in the respective partial exhaust passageways 3 a, 3 b, 3 c,3 d as follows:

Of the partial exhaust passageways 3 a, 3 b, 3 c, 3 d, each of thepartial exhaust passageways 3 a, 3 b is defined by the exhaust pipe 6 a.In order to estimate the temperatures of the exhaust gas in the partialexhaust passageways 3 a, 3 b, the temperature changes depending on thespeed of the exhaust gas and the temperature gradient thereof (thetemperature gradient in the direction in which the exhaust gas flows),the heat transfer between the exhaust gas and the exhaust pipe 6 a, andthe heat radiation from the exhaust pipe 6 a into the atmosphere aretaken into account in the same manner as described above with respect tothe circular tube 23.

An estimated value of the exhaust gas temperature Tga in the partialexhaust passageway 3 a and an estimated value of the temperature Twa(hereinafter referred to as “exhaust pipe temperature Twa”) of theexhaust pipe 6 a in the partial exhaust passageway 3 a are determined byrespective model equations (5-1), (5-2), shown below, in each cycle timeof the processing sequence of the exhaust temperature observer 19. Anestimated value of the exhaust gas temperature Tgb in the partialexhaust passageway 3 b and an estimated value of the exhaust pipetemperature Twb in the partial exhaust passageway 3 b are determined byrespective model equations (6-1), (6-2), shown below, in each cycle timeof the processing sequence of the exhaust temperature observer 19. Morespecifically, the exhaust gas temperature Tga and the exhaust pipetemperature Twa that are determined by the equations (5-1), (5-2)represent estimated values of the temperatures in the vicinity of thedownstream end of the partial exhaust passageway 3 a. Likewise, theexhaust gas temperature Tgb and the exhaust pipe temperature Twb thatare determined by the equations (6-1), (6-2) represent estimated valuesof the temperatures in the vicinity of the downstream end of the partialexhaust passageway 3 b.

$\begin{matrix}{{{Tga}\left( {k + 1} \right)} = {{{Tga}(k)} - {{Vg} \cdot \frac{dt}{La} \cdot \left( {{{Tga}(k)} - {{Texg}(k)}} \right)} + {{Aa} \cdot {dt} \cdot \left( {{{Twa}(k)} - {{Tga}(k)}} \right)}}} & \left( {5\text{-}1} \right) \\{{{Twa}\left( {k + 1} \right)} = {{{Twa}(k)} + {{Ba} \cdot {dt} \cdot \left( {{{Tga}(k)} - {{Twa}(k)}} \right)} + {{Ca} \cdot {dt} \cdot \left( {{T_{A}(k)} - {{Twa}(k)}} \right)}}} & \left( {5\text{-}2} \right) \\{{{Tgb}\left( {k + 1} \right)} = {{{Tgb}(k)} - {{Vg} \cdot \frac{dt}{Lb} \cdot \left( {{{Tgb}(k)} - {{Tga}(k)}} \right)} + {{Ab} \cdot {dt} \cdot \left( {{{Twb}(k)} - {{Tgb}(k)}} \right)}}} & \left( {6\text{-}1} \right) \\{{{Twb}\left( {k + 1} \right)} = {{{Twb}(k)} + {{Bb} \cdot {dt} \cdot \left( {{{Tgb}(k)} - {{Twb}(k)}} \right)} + {{Cb} \cdot {dt} \cdot \left( {{T_{A}(k)} - {{Twb}(k)}} \right)}}} & \left( {6\text{-}2} \right)\end{matrix}$

In the equations (5-1), (5-2), (6-1), (6-2), dt represents the period(cycle time) of the processing sequence of the exhaust temperatureobserver 19, and corresponds to Δt in the equations (4-1), (4-2). In theequations (5-1), (6-1), La, Lb represent the respective lengths (fixedvalues) of the partial exhaust passageways 3 a, 3 b, and correspond toΔz in the equation (4-1). Aa, Ba, Ca in the equations (5-1), (5-2) andAb, Bb, Cb in the equations (6-1), (6-2) represent model coefficientscorresponding respectively to a, b, c in the equations (4-1), (4-2), andthe values of those model coefficients are set (identified) in advanceby way of experimentation or simulation. In the equations (5-1), (6-1),Vg represents a parameter (to be determined as described later on)indicative of the speed of the exhaust gas, and corresponds to V in theequation (4-1).

The exhaust gas temperature Texg(k) (the exhaust gas temperature at theexhaust port 2) which is required to calculate a new estimated valueTga(k+1) of the exhaust gas temperature Tga according to the equation(5-1) is basically of the latest value determined according to theequation (1). Similarly, the exhaust gas temperature Tga(k) (the exhaustgas temperature in the partial exhaust passageway 3 a) which is requiredto calculate a new estimated value Tgb(k+1) of the exhaust gastemperature Tgb according to the equation (6-1) is basically of thelatest value determined according to the equation (5-1). The atmospherictemperature TA(k) which is required in the calculation of the equations(5-2), (6-2) is of the latest value of the atmospheric temperaturedetected by an atmospheric temperature sensor (in the presentembodiment, a sensor on the engine 1 is used for this atmospherictemperature sensor), not shown. In the present embodiment, the gas speedparameter Vg which is required in the calculation of the equations(5-1), (6-1) is of a value which is calculated from latest detectedvalues of the rotational speed NE and the intake pressure PB accordingto the following equation (7):

$\begin{matrix}{{Vg} = {\frac{NE}{NEBASE} \cdot \frac{PB}{PBBASE}}} & (7)\end{matrix}$where NEBASE, PBBASE represent a predetermined rotational speed and apredetermined intake pressure, which are set to, for example, themaximum rotational speed of the engine 1 and 760 mmHg (≈101 kPa),respectively. The gas speed parameter Vg calculated according to theequation (7) is proportional to the speed of the exhaust gas, with Vg≦1.

In the present embodiment, initial values Tga(0), Twa(0), Tgb(0), Twb(0)of the estimated values for the exhaust gas temperature Tga, the exhaustpipe temperature Twa, the exhaust gas temperature Tgb, and the exhaustpipe temperature Twb are set to the atmospheric temperature TA which isdetected by the atmospheric temperature sensor (not shown) or the enginetemperature TW (the temperature of the coolant of the engine 1) detectedby the engine temperature sensor (not shown) when the engine 1 hasstarted to operated (upon an engine startup).

The partial exhaust passageway 3 c is defined by the catalyst 7 in thecatalytic converter 4. The catalyst 7 generates heat by itself due toits own exhaust gas purifying action (specifically, anoxidizing/reducing action), and the amount of heat (the amount of heatper unit time) generated by the catalyst 7 is substantially inproportion to the speed of the exhaust gas. This is because as the speedof the exhaust gas is higher, the exhaust gas components reacting withthe catalyst 7 per unit time increase.

According to the present embodiment, for estimating the exhaust gastemperature in the partial exhaust passageway 3 c with high accuracy,the generation of heat by the catalyst 7 in the catalytic converter 4 aswell as the temperature change depending on the speed and temperaturegradient of the exhaust gas, the heat transfer between the exhaust gasand the catalyst 7, and the heat radiation from the catalyst 7 into theatmosphere are taken into account.

An estimated value of the exhaust gas temperature Tgc in the partialexhaust passageway 3 c and an estimated value of the temperature Twc(hereinafter referred to as “catalyst temperature Twc”) of the catalyst7 which defines the partial exhaust passageway 3 c are determined byrespective model equations (8-1), (8-2), shown below, in each cycle timeof the processing sequence of the exhaust temperature observer 19. Morespecifically, the exhaust gas temperature Tgc and the catalysttemperature Twc that are determined by the equations (8-1), (8-2)represent estimated values of the temperatures in the vicinity of thedownstream end of the partial exhaust passageway 3 c, i.e., in thevicinity of the outlet of the catalytic converter 4.

$\begin{matrix}{{{Tgc}\left( {k + 1} \right)} = {{{Tgc}(k)} - {{Vg} \cdot \frac{dt}{Lc} \cdot \left( {{{Tgc}(k)} - {{Tgb}(k)}} \right)} + {{Ac} \cdot {dt} \cdot \left( {{{Twc}(k)} - {{Tgc}(k)}} \right)}}} & \left( {8\text{-}1} \right) \\{{{Twc}\left( {k + 1} \right)} = {{{Twc}(k)} + {{Bc} \cdot {dt} \cdot \left( {{{Tgc}(k)} - {{Twc}(k)}} \right)} + {{Cc} \cdot {dt} \cdot \left( {{T_{A}(k)} - {{Twc}(k)}} \right)} + {{Dc} \cdot {dt} \cdot {Vg}}}} & \left( {8\text{-}2} \right)\end{matrix}$

In the equation (8-1), Lc represents the length (fixed value) of thepartial exhaust passageway 3 c, and corresponds to Δz in the equation(4-1). Ac, Bc, Cc in the equations (8-1), (8-2) represent modelcoefficients corresponding respectively to a, b, c in the equations(4-1), (4-2), and the values of those model coefficients are set(identified) in advance by way of experimentation or simulation. Thefourth term on the right side of the equation (8-2) represents atemperature change component of the catalyst 7 in the catalyticconverter 4 due to the heating of the catalyst 7 by itself, i.e., thetemperature change per period of the processing sequence of the exhausttemperature observer 19, and is proportional to the gas speed parameterVg. As with Ac through Cc, Dc in the fourth term represents a modelcoefficient that is set (identified) in advance by way ofexperimentation or simulation. Therefore, the equation (8-2) correspondsto the combination of the right side of the equation (4-2) with atemperature change component due to the heating of a passage-definingmember (the catalyst 7).

dt, Vg in the equations (8-1), (8-2) have the same meanings and valuesas those in the equations (5-1) through (6-2). The value of TA used inthe calculation of the equation (8-2) is identical to those used in theequation (5-2), (6-2). In the present embodiment, the initial valuesTgc(0), Twc(0) of the exhaust gas temperature Tgc and the catalysttemperature Twc are equal to the detected value of the atmospherictemperature TA or the detected value of the engine temperature TW at thetime the engine 1 has started to operate, as with the equations (5-1)through (6-2).

The partial exhaust passageway 3 d is defined by the exhaust pipe 6 bsimilar to the exhaust pipe 6 a which define the partial exhaustpassageways 3 a, 3 b. The exhaust gas temperature Tgd in the partialexhaust passageway 3 d and the exhaust pipe temperature Twd of theexhaust pipe 6 b, or more specifically the temperature at the downstreamend of the partial exhaust passageway 3 d, are determined respectivelyby the following equations (9-1), (9-2) which are similar to theequations (5-1) through (6-2):

$\begin{matrix}{{{Tgd}\left( {k + 1} \right)} = {{{Tgd}(k)} - {{Vg} \cdot \frac{dt}{Ld} \cdot \left( {{{Tgd}(k)} - {{Tgc}(k)}} \right)} + {{Ad} \cdot {dt} \cdot \left( {{{Twd}(k)} - {{Tgd}(k)}} \right)}}} & \left( {9\text{-}1} \right) \\{{{Twd}\left( {k + 1} \right)} = {{{Twd}(k)} + {{Bd} \cdot {dt} \cdot \left( {{{Tgd}(k)} - {{Twd}(k)}} \right)} + {{Cd} \cdot {dt} \cdot \left( {{T_{A}(k)} - {{Twd}(k)}} \right)}}} & \left( {9\text{-}2} \right)\end{matrix}$

In the equation (9-1), Ld represents the length (fixed value) of thepartial exhaust passageway 3 d, and corresponds to Δz in the equation(4-1). Ad, Bd, Cd in the equations (9-1), (9-2) represent modelcoefficients corresponding respectively to a, b, c in the equations(4-1), (4-2), and the values of those model coefficients are set(identified) in advance by way of experimentation or simulation.

dt, Vg in the equations (9-1), (9-2) have the same meanings and valuesas those in the equations (5-1) through (6-2). The value of TA used inthe calculation of the equation (9-2) is identical to those used in theequation (5-2), (6-2), (8-2). The initial values Tgd(0), Twd(0) of theexhaust gas temperature Tgd and the catalyst temperature Twd are equalto the detected value of the atmospheric temperature TA or the detectedvalue of the engine temperature TW at the time the engine 1 has startedto operate, as with the equations (5-1) through (6-2).

The processing sequence of the exhaust temperature observer 19, asdescribed above, determines estimated values of the exhaust gastemperatures Texe, Tga, Tgb, Tgc, Tgd in the exhaust port 2 of theengine 1 and the partial exhaust passageways 3 a, 3 b, 3 c, 3 dsuccessively downstream in each cycle time. The estimated value of theexhaust gas temperature Tgd in the partial exhaust passageway 3 d whichis located most downstream corresponds to the temperature of the exhaustgas in the vicinity of the location of the O₂ sensor 8. The estimatedvalue of the exhaust gas temperature Tgd is obtained as the estimatedvalue of the exhaust gas temperature in the vicinity of the location ofthe O₂ sensor 8. In the present embodiment, the estimated value of theexhaust gas temperature Tgd corresponds to exhaust gas temperature dataaccording to the present invention.

The algorithm of the estimating process of the exhaust temperatureobserver 19 is shown in block form in FIG. 6. In FIG. 6, the modelequation (1) is referred to as an exhaust port thermal model 24, themodel equations (5-1), (5-2) and the model equations (6-1), (6-2) aspre-CAT exhaust system thermal models 25, 26, respectively, the modelequations (8-1), (8-2) as an in-CAT exhaust system thermal model 27, andthe model equations (9-1), (9-2) as a post-CAT exhaust system thermalmodel 28. As shown in FIG. 6, each of the thermal models 24 through 28is supplied with the detected values of the rotational speed NE and theintake pressure PB of the engine 1. The detected values of therotational speed NE and the intake pressure PB which are supplied to theexhaust port thermal model 24 are used to determine the basic exhaustgas temperature TMAP, and the detected values of the rotational speed NEand the intake pressure PB which are supplied to the exhaust systemthermal models 25 through 28 are used to determine the value of the gasspeed parameter Vg. Each of the thermal models 25 through 28 is alsosupplied with the detected value of the atmospheric temperature TA. Thepre-CAT exhaust system thermal model 25, the pre-CAT exhaust systemthermal model 26, the in-CAT exhaust system thermal model 27, and thepost-CAT exhaust system thermal model 28 are supplied with the estimatedvalues of the exhaust gas temperatures Texg, Tga, Tgb, Tgc,respectively, which are outputted from the higher-level thermal models24, 25, 26, 27. The post-CAT exhaust system thermal model 28 eventuallyproduces the estimated value of the exhaust gas temperature Tgd in thevicinity of the location of the O₂ sensor 8.

In the present embodiment, the detected value produced by theatmospheric temperature sensor on the engine 1 is used to estimate thetemperatures of the passage-defining members (the exhaust pipe 6 a, thecatalyst 7 in the catalytic converter 4, and the exhaust pipe 6 b) whichdefine the partial exhaust passageways 3 a, 3 b, 3 c, 3 d. However, thedetected value produced by an atmospheric sensor which is disposedoutside of the exhaust passage 3 may be used to estimate thetemperatures of those passage-defining members.

The element temperature observer 20 will be described below. The elementtemperature observer 20 estimates the temperature T_(O2) of the activeelement 10 of the O₂ sensor 8 sequentially in given cycle times in viewof the thermal transfer between the active element 10 and the exhaustgas held in contact therewith, the heat radiation from the activeelement 10 into the air within the active element 10, and the thermaltransfer between the active element 10 and the ceramic heater 13(hereinafter referred to simply as “heater 13”) which heats the activeelement 10. The element temperature observer 20 also estimates thetemperature Tht of the heater 13 in order to estimate the temperatureT_(O2) of the active element 10. In estimating the temperature Tht ofthe heater 13, the element temperature observer 20 takes into accountthe heat transfer between the heater 13 and the active element 13 andthe heat radiation from the heater 13 into the air within the activeelement 10, and also the heating of the heater 13 based on the electricenergy supplied to the heater 13. The element temperature observer 20has an estimating algorithm for estimating the temperature T_(O2) andthe temperature Tht, which is constructed as follows:

The element temperature observer 20 determines an estimated value of thetemperature T_(O2) of the active element 10 (hereinafter referred to as“element temperature T_(O2)”) of the O₂ sensor 8 and an estimated valueof the temperature Tht of the heater 13 (hereinafter referred to as“heater temperature Tht”) sequentially in given cycle times respectivelyaccording to the model equations (10-1), (10-2) described below. Theequation (10-1) is the equation of an element temperature model, and theequation (10-2) is the equation of a heater temperature model.

$\begin{matrix}{{T_{O\; 2}\left( {k + 1} \right)} = {{T_{O\; 2}(k)} + {{Ax} \cdot {dt} \cdot \left( {{{Tgd}(k)} - {T_{O\; 2}(k)}} \right)} + {{Bx} \cdot {dt} \cdot \left( {{{Tht}(k)} - {T_{O\; 2}(k)}} \right)} - {{Ex} \cdot {dt} \cdot \left( {{T_{O\; 2}(k)} - {{TA}^{\prime\;}(k)}} \right)}}} & \left( {10\text{-}1} \right) \\{{{Tht}\left( {k + 1} \right)} = {{{Tht}(k)} - {{Cx} \cdot {dt} \cdot \left( {{{Tht}(k)} - {T_{O\; 2}(k)}} \right)} - {{Fx} \cdot {dt} \cdot \left( {{{Tht}(k)} - {{TA}^{\prime}(k)}} \right)} + {{Dx} \cdot {dt} \cdot {{DUT}(k)} \cdot \frac{{{VB}(k)}^{2}}{{NVB}^{2}}}}} & \left( {10\text{-}2} \right)\end{matrix}$

The equation (10-1) indicates that the temperature change of the activeelement 10 in each cycle time depends on a temperature change component(the second term on the right side of the equation (10-1)) depending onthe difference between the exhaust gas temperature Tgd in the vicinityof the location of the O₂ sensor 8 (the exhaust gas temperature in thepartial exhaust passageway 3 d) and the element temperature T_(O2),i.e., a temperature change component which is caused by the heattransfer between the active element 10 and the exhaust gas held incontact therewith, a temperature change component (the third term on theright hand of the equation (10-1)) depending on the difference betweenthe element temperature T_(O2) and the heater temperature Tht, i.e., atemperature change component which is caused by the heat transferbetween the active element 10 and the ceramic heater 13, and atemperature change component (the fourth term on the right hand of theequation (10-1)) depending on the difference between the elementtemperature T_(O2) and the temperature TA′ of the air in the activeelement 10, i.e., a temperature change component due to the heatradiation from the active element 10 into the air therein, i.e., the sumof those temperature change components.

The equation (10-2) indicates that the temperature change of the heater13 in each cycle time depends on a temperature change component (thesecond term on the right side of the equation (10-2)) depending on thedifference between the element temperature T_(O2) and the heatertemperature Tht, i.e., a temperature change component which is caused bythe heat transfer between the active element 10 and the heater 13, atemperature change component (the third term on the right hand of theequation (10-2)) depending on the difference between the heatertemperature Tht and the temperature TA′ of the air in the active element10, i.e., a temperature change component due to the heat radiation fromthe heater 13 into the air within the active element 10, and atemperature change component (the fourth term on the right side of theequation (10-2)) depending on the product of the duty cycle DUT (moreaccurately, the duty cycle DUT that is actually used for the heatercontroller 22 to control the energization of the heater 13) that isgenerated by the heat controller 22 as described later on and the squareVB² of the battery voltage VB, i.e., a temperature change componentwhich is caused by the heating of the heater 13 based on the electricenergy supplied thereto, i.e., the sum of those temperature changecomponents.

In the equations (10-1), (10-2), Ax, Bx, Cx, Dx, Ex, Fx represent modelcoefficients whose values are set (identified) in advance by way ofexperimentation or simulation, and dt represents the period (cycle time)of the processing sequence of the element temperature observer 20. Inthe present embodiment, the period dt is set to the same value as thecycle time (represented by dt in the equations (5-1) through (9-2)) ofthe processing sequence of the exhaust temperature observer 19. In theequation (10-2), NVB represents a predetermined reference value (e.g.,14 V) of the battery voltage VB. The reference value may basically beset as desired as a standard voltage (a voltage that can normally beemployed) for the battery voltage VB.

The fourth term on the right side of the equation (10-2) willsupplementarily be described below. If the duty cycle of the PWM controlfor the heater 13 is constant and the resistance of the heater 13 uponits energization is constant, then the electric power supplied to theheater 13 is proportional to the square of the voltage applied to theheater 13, and the applied voltage is proportional to the batteryvoltage VB. The duty cycle DUT defines the period of time in which theheater 13 is energized, per period of the pulse voltage for the PWMcontrol. Therefore, the product of the duty cycle DUT and the square VB²of the battery voltage VB is proportional to the electric power suppliedto the heater 13. The battery voltage VB changes as an alternator forcharging the battery is turned on and off, for example. In the equation(10-2), the duty cycle DUT and the square VB² of the battery voltage VBare multiplied in order to obtain a temperature change component due tothe heating of the heater 13 when it is supplied with the electricpower.

The duty cycle DUT(k) which is required in the calculation of theequation (10-2) is of the latest value of the duty cycle DUT that isactually used for the heater 22 to control the energization of theheater 13 (PWM control). In the present embodiment, the latest value ofthe atmospheric temperature TA detected by the atmospheric temperaturesensor is substituted for the temperature TA′(k) of the air in theactive element 10 which is required in the calculation of the equations(10-1), (10-2). In the present embodiment, therefore, TA′(k)=TA(k).Furthermore, the initial values T_(O2)(0), Tht(0) of the elementtemperature TO₂ and the heater temperature Tht are equal to the detectedvalue of the atmospheric temperature TA or the detected value of theengine temperature TW at the time the engine 1 has started to operate,as described later.

The element temperature observer 20 sequentially calculates theestimated values of the element temperature T_(O2) and the heatertemperature Tht according to the estimating algorithm described above.The estimated values of the element temperature T_(O2) and the heatertemperature Tht correspond respectively to element temperature data andheater temperature data in the present invention.

The heater controller 22 will be described below. Basically, the heatercontroller 22 sequentially generates the duty cycle DUT as a controlinput (manipulated variable) for controlling the heater 13 according toan optimum predictive control algorithm, and controls the electricenergy supplied to the heater 13 with the generated duty cycle DUT.

According to the present embodiment, attention is paid to the differencebetween the element temperature T_(O2) and a target value therefor, achange per given time in the difference (corresponding to a rate ofchange of the difference), and a change per given time in the heatertemperature Tht (corresponding to a rate of change of the heatertemperature Tht), and model equations for an object to be controlled bythe heater controller 22 are introduced using the above differences andchanges as state quantities relative to the object to be controlled bythe heater controller 22. The heater controller 22 has its algorithm(optimum predictive control algorithm) constructed as described below.In the present embodiment, a duty cycle (control input) generated by theoptimum predictive control algorithm, to be described below, is moreprecisely a duty cycle required to control the element temperatureT_(O2) at a target value providing the battery voltage VB is keptconstant at the reference value NVB represented by the equation (10-2).The duty cycle generated by the optimum predictive control algorithmwhich will be described below is referred to as a basic duty cycle SDUTor a basic control input SDUT.

First, model equations for the object to be controlled by the heatercontroller 22 will be described below. Changes ΔT_(O2), ΔTht per giventime in the element temperature T_(O2) and the heater temperature Thtare expressed by the following equations (11-1), (11-2) based on therespective model equations (10-1), (10-2) with respect to the elementtemperature observer 20. The equation (11-2) is derived from an equationproduced by using VB(k)=NVB and replacing DUT with SDUT in the equation(10-2). For deriving the equations (11-1), (11-2), the temperature TA′in the active element 10 is kept constant, i.e., remains unchanged,i.e., TA′(k+1)=TA′(k), in at least one cycle time of the processingsequence of the heater controller 22.

$\begin{matrix}\begin{matrix}{{\Delta\;{T_{O\; 2}\left( {k + 1} \right)}} =} & {{\Delta\;{T_{O\; 2}(k)}} + {{Ax} \cdot {dt} \cdot \left( {{\Delta\;{{Tgd}(k)}} - {\Delta\;{T_{O\; 2}(k)}}} \right)} +} \\ & {{{Bx} \cdot {dt} \cdot \left( {{\Delta\;{{Tgt}(k)}} - {\Delta\;{T_{O\; 2}(k)}}} \right)} - {{Ex} \cdot {dt} \cdot}} \\ & {\Delta\;{T_{O\; 2}(k)}} \\{=} & {{{\left( {1 - {{Ax} \cdot {dt}} - {{Bx} \cdot {dt}} - {{Ex} \cdot {dt}}} \right) \cdot \Delta}\;{T_{O\; 2}(k)}} +} \\ & {{{{Ax} \cdot {dt} \cdot \Delta}\;{{Tgd}(k)}} + {{{Bx} \cdot {dt} \cdot \Delta}\;{{Tht}(k)}}}\end{matrix} & \left( {11\text{-}1} \right) \\\begin{matrix}{{\Delta\;{{Tht}\left( {k + 1} \right)}} =} & {{\Delta\;{{Tht}(k)}} - {{Cx} \cdot {dt} \cdot \left( {{\Delta\;{{Tht}(k)}} - {\Delta\;{T_{O\; 2}(k)}}} \right)} -} \\ & {{{{Fx} \cdot {dt} \cdot \Delta}\;{{Tht}(k)}} + {{{Dx} \cdot {dt} \cdot \Delta}\;{{SDUT}(k)}}} \\{=} & {{{\left( {1 - {{Cx} \cdot {dt} \cdot {Fx} \cdot {dt}}} \right) \cdot \Delta}\;{{Tht}(k)}} + {{Cx} \cdot {dt} \cdot}} \\ & {{\Delta\;{T_{O\; 2}(k)}} + {{{Dx} \cdot {dt} \cdot \Delta}\;{{SDUT}(k)}}}\end{matrix} & \left( {11\text{-}2} \right)\end{matrix}$

In the above equations (11-1), (11-2), ΔT_(O2)(k)=T_(O2)(k+1)−T_(O2)(k),ΔTht(k)=Tht(k+1)−Tht(k), ΔTgd(k)=Tgd(k+1)−Tgd(k),ΔSDUT(k)=SDUT(k+1)−SDUT(k).

A target value for the element temperature T_(O2) is represented by R,and the difference e between the element temperature T_(O2) and thetarget value R, i.e., the difference in each cycle time (hereinafterreferred to as “element temperature difference e”), is defined accordingto the following equation (12):e(k)=T _(O2)(k)−R(k)  (12)

A change Δe in the element temperature difference e in each cycle time(hereinafter referred to as “element temperature difference change Δe”)is expressed by the following equation (13) based on the above equations(11-1), (12):

$\begin{matrix}\begin{matrix}{{\Delta\;{e\left( {k + 1} \right)}} =} & {{\Delta\;{T_{O\; 2}\left( {k + 1} \right)}} - {\Delta\;{R\left( {k + 1} \right)}}} \\{=} & {{{\left( {1 - {{Ax} \cdot {dt}} - {{Bx} \cdot {dt}} - {{Ex} \cdot {dt}}} \right) \cdot \Delta}\;{e(k)}} + {{Ax} \cdot {dt} \cdot}} \\ & {{\Delta\;{{Tgd}(k)}} + {{{Bx} \cdot {dt} \cdot \Delta}\;{{Tht}(k)}} - {\Delta\;{R\left( {k + 1} \right)}} +} \\ & {{\left( {1 - {{Ax} \cdot {dt}} - {{Bx} \cdot {dt}} - {{Ex} \cdot {dt}}} \right) \cdot \;\Delta}\;{R(k)}}\end{matrix} & (13)\end{matrix}$

In the equation (13), Δe(k)=e(k+1)−e(k), ΔR(k)=R(k+1)−R(k). In derivingthe equation (13), the equation ΔT_(O2)=Δe(k)+ΔR(k) (based on theequation (12)) is employed.

The equation ΔT_(O2)=Δe(k)+ΔR(k) is applied to the equation (11-2), andthe resulting equation is modified into the following equation (14):ΔTht(k+1)=(1−Cx·dt−Fx·dt)·ΔTht(k)+Cx·dt·Δe(k)+Dx·dt·ΔSDUT(k)+Cx·dt·ΔR(k)  (14)

If a state quantity vector X0(k)=(e(k),Δe(k), ΔTht(k))^(T) (T representsa transposition) is introduced, then the following equation (15) isobtained from the equations (13), (14) and the equatione(k+1)=e(k)+Δe(k):X0(k+1)=Φ·X0(k)+G·ΔSDUT(k)+Gd·ΔTgd(k)+Gr·R0(k+1)  (15)

${{{where}\mspace{14mu} X\; 0(k)} = \left( {{e(k)},{\Delta\;{e(k)}},{\Delta\;{{Tht}(k)}}} \right)^{T}},{{R\; 0\left( {k + 1} \right)} = \left( {{\Delta\;{R\left( {k + 1} \right)}},{\Delta\;{R(k)}}} \right)^{T}},{G = \left( {0,0,{{Dx} \cdot {dt}}} \right)^{T}},{{Gd} = \left( {0,{{Ax} \cdot {dt}},0} \right)^{T}},{\Phi = {{\begin{bmatrix}1 & 1 & 0 \\0 & {1 - {{Ax} \cdot {dt}} - {{Bx} \cdot {dt}} - {{Ex} \cdot {dt}}} & {{Bx} \cdot {dt}} \\0 & {{Cx} \cdot {dt}} & {1 - {{Cx} \cdot {dt}} - {{Fx} \cdot {dt}}}\end{bmatrix}{Gr}} = \begin{bmatrix}0 & 0 \\{- 1} & {1 - {{Ax} \cdot {dt}} - {{Bx} \cdot {dt}} - {{Ex} \cdot {dt}}} \\0 & {{Cx} \cdot {dt}}\end{bmatrix}}}$In the equation (15), R0, G, Gd represent vectors defined in the abovedefinition clause, and Φ, Gr represent matrixes defined in the abovedefinition clause.

The above equation (15) is a basic equation of the model of the objectto be controlled by the heater controller 22.

In the above description, the period of the control process of theheater controller 22 is the same as the period dt of the processingsequences of the exhaust temperature observer 19 and the elementtemperature observer 20. Therefore, the period dt is used in the vectorsG, Gd and the matrixes Φ, Gr in the equation (15). It is preferable tocarry out the processing sequences of the exhaust temperature observer19 and the element temperature observer 20 in a relatively short period(e.g., a period of 20 through 50 msec.) in order to increase theaccuracy with which to estimate the temperatures. However, the period ofthe control process of the heater controller 22 may be longer than theperiod dt of the processing sequences of the exhaust temperatureobserver 19 and the element temperature observer 20 because the responsespeed of a change in the element temperature in response to the controlinput (duty cycle DUT) is relatively low (several Hz in terms offrequencies). According to an optimum predictive control process to bedescribed later on, since future values of the target value R of theelement temperature T_(O2) need to be stored and held for a certaintime, the storage capacity of a memory for storing the target value Rbecomes large if the period of the control process of the heatercontroller 22 is short.

According to the present embodiment, the period (cycle time) of thecontrol process of the heater controller 22 is set to a value dtc (e.g.,300 through 500 msec.) longer than the period dt of the processingsequences of the exhaust temperature observer 19 and the elementtemperature observer 20.

In the present embodiment, the model equation of the object to becontrolled by the heater controller 22 is rewritten from the equation(15) into the following equation (16), using the period dtc of thecontrol process of the heater controller 22:X0(n+1)=Φ·X0(n)+G·ΔSDUT(n)+Gd·ΔTgd(n)+Gr·R0(n+1)  (16)

${{{where}\mspace{14mu} X\; 0(n)} = \left( {{e(n)},{\Delta\;{e(n)}},{\Delta\;{{Tht}(n)}}} \right)^{T}},{{R\; 0\left( {n + 1} \right)} = \left( {{\Delta\;{R\left( {n + 1} \right)}},{\Delta\;{R(n)}}} \right)^{T}},{G = \left( {0,0,{{Dx} \cdot {dtc}}} \right)^{T}},{{Gd} = \left( {0,{{Ax} \cdot {dtc}},0} \right)^{T}},{\Phi = \begin{bmatrix}1 & 1 & 0 \\0 & {1 - {{Ax} \cdot {dtc}} - {{Bx} \cdot {dtc}} - {{Ex} \cdot {dtc}}} & {{Bx} \cdot {dtc}} \\0 & {{Cx} \cdot {dtc}} & {1 - {{Cx} \cdot {dtc}} - {{Fx} \cdot {dtc}}}\end{bmatrix}}$ ${Gr} = \begin{bmatrix}0 & 0 \\{- 1} & {1 - {{Ax} \cdot {dtc}} - {{Bx} \cdot {dtc}} - {{Ex} \cdot {dtc}}} \\0 & {{Cx} \cdot {dtc}}\end{bmatrix}$

The equation (16) is a model equation of the object to be controlledwhich is actually used in the algorithm of the control process of theheater controller 22. In the equation (16), n represents the ordinalnumber of the period dtc of the control process of the heater controller22.

Using the above model equation, the algorithm of the control process ofthe heater controller 22, i.e., the algorithm of the optimum predictivecontrol process, is constructed as follows: It is assumed that thetarget value R of the element temperature T_(O2) is set for the futureuntil after Mr steps (until after a multiple by Mr of the period dtc ofthe control process of the heater controller 22), and the exhaust gastemperature Tgd which acts as a disturbance input is known in the futureuntil after Md steps (until after a multiple by Md of the period dtc ofthe control process of the heater controller 22). The value Mr will bereferred to as a target value predicting time Mr, and the value Md as anexhaust gas temperature predicting time Md. These predicting times Mr,Md are represented by integers whose unit is one period dtc of thecontrol process of the heater controller 22.

A controller for generating a control input ΔSDUT for minimizing thevalue of an evaluating function J0 according to the following equation(17) serves as an optimum predictive servo controller:

$\begin{matrix}{{J\; 0} = {\sum\limits_{n = {M + 1}}^{\infty}\;\left\lbrack {{X\; 0^{T}{(n) \cdot Q}\;{0 \cdot X}\; 0(n)} + {\Delta\;{{{SDUT}^{T}(n)} \cdot H}\;{0 \cdot \Delta}\;{{SDUT}(n)}}} \right\rbrack}} & (72)\end{matrix}$where M represents a larger one of the target value predicting time Mrand the exhaust gas temperature predicting time Md, i.e., M=max(Mr,Md),and Q0, H0 weighted matrixes for adjusting the convergence of the statequantity vector X0 and the power (size) of the control input ΔSDUT. Q0represents a 3-row, 3-column diagonal matrix as X0 is a cubic matrix,and H0 is a Scalar quantity as ΔSDUT is a Scalar quantity. In thepresent embodiment, in order to reduce the power consumption of theheater 13, Q0 is set to a unit matrix (a diagonal matrix whose alldiagonal components are “1”) and H0 is set to a value (e.g., 1000)greater than the diagonal components of the matrix Q0. The target valuepredicting time Mr is set to 20, for example, and the exhaust gastemperature predicting time Md is set to 10, for example, with theperiod of the control process of the heater controller 22 being in therange from 300 to 500 msec.

The control input ΔSDUT for minimizing the value of the evaluatingfunction according to the equation (17) is expressed by the equation(18) given below. In the present embodiment, it is assumed that theexhaust gas temperature Tgd is maintained at the present value in thefuture until after Md steps.

$\begin{matrix}{{\Delta\;{{SDUT}(n)}} = {{F\;{0 \cdot X}\; 0(n)} + {\sum\limits_{i = 0}^{Mr}\;\left\lbrack {{Fr}\; 0{(i) \cdot R}\; 0\left( {n + 1} \right)} \right\rbrack} + {{{Fdt} \cdot \Delta}\;{{Tgd}(n)}}}} & (18)\end{matrix}$

In the equation (18), F0 in the first term on the right side representsa cubic row vector (Fs0,Fe0,Fx0), Fr0(i) (i=1, 2, . . . , Mr) in thesecond term (the term of Σ) on the right side represent quadratic rowvectors (Fr01(i), Fr02(i), and Fdt in the third term on the right siderepresents a Scalar quantity. They are expressed by the equations (19-1)through (19-3) given below.

$\begin{matrix}\begin{matrix}{{F\; 0} \equiv \left( {{{Fs}\; 0},{{Fe}\; 0},{{Fx}\; 0}} \right)} \\{= {{- \left\lbrack {{H\; 0} + {G^{T} \cdot P \cdot G}} \right\rbrack^{- 1}} \cdot G^{T} \cdot P \cdot \Phi}}\end{matrix} & \left( {19\text{-}1} \right) \\\begin{matrix}{{{Fr}\; 0(i)} \equiv {\left( {{{Fr}\; 01(i)},{{Fr}\; 02(i)}} \right)\mspace{14mu}\left( {{i = 1},2,\cdots,{Mr}} \right)}} \\{= {{- \left\lbrack {{H\; 0} + {G^{T} \cdot P \cdot G}} \right\rbrack^{- 1}} \cdot G^{T} \cdot \left( \zeta^{T} \right)^{i - 1} \cdot P \cdot {Gr}}}\end{matrix} & \left( {19\text{-}2} \right) \\{{Fdt} = {\sum\limits_{i = 0}^{Md}\;\left\{ {{- \left\lbrack {{H\; 0} + {G^{T} \cdot P \cdot G}} \right\rbrack^{- 1}} \cdot G^{T} \cdot \left( \zeta^{T} \right)^{i} \cdot P \cdot {Gd}} \right\}}} & \left( {19\text{-}3} \right)\end{matrix}$where P represents a matrix (a 3-row, 3-column matrix) satisfying thefollowing Ricatti equation (20-1), and ζ represents a matrix (a 3-row,3-column matrix) expressed by the following equation (20-2):P=Q0+Φ^(T) ·P·Φ−Φ·P·G·[H0+G ^(T) ·P·G] ⁻¹ ·G ^(T) ·P·Φ  (20-1)ζ=Φ+G·F0  (20-2)

G, Gr, Gd, and Φ in the equations (19-1) through (19-3) and theequations (20-1), (20-2) are defined in the definition clause for theequation (16), and H0, Q0 in those equations represent weighted matrixesof the evaluating function J0 according to the equation (17) (H0 is aScalar quantity).

The second term (the term of Σ) on the right side of the equation (18)is rewritten using the components of Fr0, R0 (see the definition clausesfor the equations (19-2), (16)), and then modified into the followingequation (21):

$\begin{matrix}{{{\sum\limits_{i = 1}^{Mr}\;\left\lbrack {{Fr}\; 0{(i) \cdot R}\; 0\left( {n + 1} \right)} \right\rbrack} = {\sum\limits_{i = 1}^{Mr}\;\left\lbrack {{{{Fr}(i)} \cdot \Delta}\;{R\left( {n + 1} \right)}} \right\rbrack}}{where}{{{Fr}(i)} = \left\lbrack \begin{matrix}{{Fr}\; 02(1)} & {{:i} = 0} \\{{{Fr}\; 01(i)} + {{Fr}\; 02\left( {i + 1} \right)}} & {{{:i} = 1},2,\cdots,{{Mr} - 1}} \\{{Fr}\; 01({Mr})} & {{:i} = {Mr}}\end{matrix} \right.}} & (21)\end{matrix}$

By putting the equation (21) into the equation (18) and rewriting thefirst term on the right side of the equation (18) using the componentsof F0, X0 (see the definition clauses for the equations (19-1), (16)),the equation (18) is expressed by the following equation (22):

$\begin{matrix}{{\Delta\;{{SDUT}(n)}} = {{{Fs}\;{0 \cdot {e(n)}}} + {{Fe}\;{0 \cdot \Delta}\;{e(n)}} + {{Fx}\;{0 \cdot \Delta}\;{{Tht}(n)}} + {\sum\limits_{i = 0}^{Mr}\;\left\lbrack {{{{Fr}(i)} \cdot \Delta}\;{R\left( {n + 1} \right)}} \right\rbrack} + {{{Fdt} \cdot \Delta}\;{{Tgd}(n)}}}} & (22)\end{matrix}$

Since the basic control input SDUT(n) to be generated by the heatercontroller 22 is represented by the sum of its initial value SDUT(0) andΔSDUT(1), ΔSDUT(2), . . . , ΔSDUT(n) cumulatively added thereto, thefollowing equation (23) is obtained from the above equation (22):

$\begin{matrix}{{{SDUT}(n)} = {{{Fs}\;{0 \cdot {\sum\limits_{j = 1}^{n}\;{e(j)}}}} + {{Fe}\;{0 \cdot {e(n)}}} + {{Fx}\;{0 \cdot {{Tht}(n)}}} + {\sum\limits_{i = 0}^{Mr}\;\left\lbrack {{{Fr}(i)} \cdot {R\left( {n + i} \right)}} \right\rbrack} + {{Fdt} \cdot {{Tgd}(n)}} - {{Fe}\;{0 \cdot {e(0)}}} - {{Fx}\;{0 \cdot {{Tht}(0)}}} - {\sum\limits_{i = 0}^{Mr}\;\left\lbrack {{{Fr}(i)} \cdot {R\left( {0 + i} \right)}} \right\rbrack} - {{Fdt} \cdot {{Tgd}(0)}} + {{SDUT}(0)}}} & (23)\end{matrix}$

By setting the initial value terms of the equation (23), i.e., the sixthterm (the term of Fe0·e(0)) through the tenth term (SDUT(0)), to “0”,the following equation (24) is obtained as an equation for calculatingthe basic control input SDUT(n) to be actually generated by the heatercontroller 22:

$\begin{matrix}{{{SDUT}(n)} = {{{Fs}\;{0 \cdot {\sum\limits_{j = 1}^{n}\;{e(j)}}}} + {{Fe}\;{0 \cdot {e(n)}}} + {{Fx}\;{0 \cdot {{Tht}(n)}}} + {\sum\limits_{i = 0}^{Mr}\;\left\lbrack {{{Fr}(i)} \cdot {R\left( {n + 1} \right)}} \right\rbrack} + {{Fdt} \cdot {{Tgd}(n)}}}} & (24)\end{matrix}$

The equation (24) is a formula for calculating the basic control inputSDUT(n) (basic duty cycle) for controlling the heater 13 with the heatercontroller 22 according to the optimum predictive control algorithm. Thefirst through third terms (the term including Σe(j) through the termincluding Tht(n)) of the equation (24) represent control inputcomponents (a feedback component which will hereinafter be referred toas “optimum F/B component Uopfb”) depending on the element temperaturedifference e and the heater temperature Tht. Specifically, the first andsecond terms represent a control input component depending on theelement temperature difference e, and the third term represents acontrol input component depending on the heater temperature Tht. Thefourth term (the term of ΣFr(i)·R(n+1)) on the right side of theequation (24) represents a control input component (a feed-forwardcomponent which will hereinafter be referred to as “optimum target valueF/F component Uopfr”) depending on the target value R. The fifth term(the term including Tgd(n)) represents a control input component (afeed-forward component which will hereinafter be referred to as “optimumdisturbance F/F component Uopfd”) depending on the exhaust gastemperature Tgd (which functions as a disturbance on the object to becontrolled).

The basic control input SDUT(n) (basic duty cycle) determined by theequation (24) is a control input (duty cycle) that is required tocontrol the element temperature T_(O2) at the target value R in the casewhere the battery voltage VB is kept constant at the reference valueSVB, as described above. Therefore, the heater controller 22sequentially calculates a control input DUT(n) which is capable ofcontrolling the element temperature T_(O2) at the target value Rindependent of the battery voltage VB by correcting the basic controlinput SDUT(n), which has been calculated in each cycle time (period) ofthe control process according to the equation (24), with the ratioNVB²/VB(n)² of the square of the present value VB(n) of the batteryvoltage VB and the square of the reference value NVB, as indicated bythe following equation (25):

$\begin{matrix}{{{DUT}(n)} = {\frac{{NVB}^{2}}{{{VB}(n)}^{2}} \cdot {{SDUT}(n)}}} & (25)\end{matrix}$

The heater controller 22 limits the control input DUT(n), i.e., the dutycycle DUT(n), within a predetermined range (0%≦DUT(n)≦100%), and thenapplies a pulse voltage having the duty cycle DUT(n) to a heaterenergization circuit (not shown) to adjust the electric power suppliedto the heater 13.

The heater controller 22 which determines the duty cycle DUT as thecontrol input according to the equations (24), (25) is expressed inblock form as shown in FIG. 7.

In the present embodiment, since the exhaust gas temperature Tgd ismaintained at the present value in the future until after Md steps, Fdtin the fifth term on the right side of the equation (24) is a Scalarquantity. If Tgd in each step in the future can be detected orestimated, then the control input DUT can be determined using thosevalues of Tgd. In this case, Fdt is a vector comprising elements ((Md+1)elements) within { } of the equation (19-3). More specifically, if theseries-series data of the exhaust gas temperature Tgd from the presentuntil after Md steps is represented by Tgd(n), Tgd(n+1), . . . ,Tgd(n+Md), then the fifth term on the right side of the equation (24) isexpressed by the inner product (Scalar product) of the vector comprisingelements ((Md+1) elements) within { } of the equation (19-3) and avector comprising, as elements, the series-series data Tgd(n), Tgd(n+1),. . . , Tgd(n+Md) of the exhaust gas temperature Tgd. The inner productis in conformity with the fifth term on the right side of the equation(24) when Tgd(n)=Tgd(n+1)= . . . =Tgd(n+Md).

Fs0, Fe0, Fx0 which are required to determine the control input DUT(n)according to the equation (24) are of values calculated in advanceaccording to the equation (19-1). Fr(i) (i=0, 1, . . . , Mr) is ofvalues calculated in advance according to the equations (21), (19-2).Fdt is of a value calculated in advance according to the equation(19-3). These coefficients Fs0, Fe0, Fx0, Fr(i), Fdt may not necessarilybe of the values according to the defining equations, but may be ofvalues adjusted by way of simulation or experimentation. Furthermore,the coefficients Fs0, Fe0, Fx0, Fr(i), Fdt may be changed depending onthe element temperature, the heater temperature, etc.

The heater temperature Tht and the exhaust gas temperature Tgd which arerequired in the calculation of the equation (24) are of the latestestimated value of the heater temperature Tht determined by the elementtemperature observer 20 and the latest estimated value of the exhaustgas temperature Tgd determined by the exhaust temperature observer 19.

The element temperature difference e required in the calculation of theequation (24) is calculated from the latest estimated value of theelement temperature T_(O2) determined by the element temperatureobserver 20 and the target value R which has been set in a cycle timeprior to the target value predicting time Mr by the target value settingmeans 21.

The target value setting means 21 basically sets a temperature (e.g.,800° C. in the present embodiment) equal to or higher than 750° C. atwhich the output characteristics of the O₂ sensor 8 are stably good, asthe target value R for the temperature of the active element 10 in thesame cycle time as the cycle time (period) of the processing sequence ofthe heater controller 22. In order to perform the processing sequence ofthe heater controller 22 according to the algorithm of the optimumpredictive control process, the target value setting means 21 sets thetarget value R in each cycle time as a target value R(n+Mr) after thetarget value predicting time Mr from the present cycle time, and storesa series of target values R(n+Mr) for the target value predicting timeMr. Specifically, the target value setting means 21 stores Mr+1 targetvalues R(n), R(n+1), . . . , R(n+Mr) while sequentially updating them.The target value R used to determine the element temperature differencee that is required in the calculation of the equation (24) is the valueR(n) set and stored by the target value setting means 21 as describedabove in the cycle time prior to the target value predicting time Mr.The target values R(n), R(n+1), . . . , R(n+Mr) stored as describedabove are used to determine the value of the fourth term (the term of Σincluding R(n+i)) of the equation (24).

If the target value R of the element temperature T_(O2) is set to a hightemperature such as 800° C. from the start of operation of the engine 1,then the active element 10 tends to be damaged due to stresses caused byquick heating if water is applied to the active element 10 of the O₂sensor 8 when the engine 1 starts to operate. In the present invention,therefore, until a certain time (e.g., 15 seconds) elapses from thestart of operation of the engine 1, the target value setting means 21sets the target value R of the element temperature T_(O2) to atemperature lower than 750° C., e.g., 600° C. In the present embodiment,if the atmospheric temperature TA is low (e.g., TA<0° C.) upon elapse ofa predetermined period of time after the engine 1 has started tooperate, the target value R for the element temperature T_(O2) is set toa temperature (750° C.≦R<800° C.) slightly lower than the normal targetvalue (800° C.) in order to prevent the heater 13 from being overheated.

Overall operation of the apparatus, particularly, the sensor temperaturecontrol means 18, according to the present embodiment will be describedbelow.

When the engine 1 starts to operate (upon an engine startup), the sensortemperature control means 18 sets initial values Texg(0), Tga(0),Tgb(0), Tgc(0), Tgd(0), Twa(0), Twb(0), Twd(0), T_(O2)(0), and Tht(0) ofthe estimated values of the exhaust gas temperatures Texg, Tga, Tgb,Tgc, Tgd, the exhaust pipe temperatures Twa, Twb, Twd, the catalysttemperature Twc, the element temperature T_(O2), and the heatertemperature Tht, as follows: In the present embodiment, while the engine1 stops its operation, the shutdown time of the engine 1 is measured.When the engine 1 starts to operate, the sensor temperature controlmeans 18 determines whether the shutdown time immediately prior to theengine startup is in excess of a predetermined time (e.g., 2 hours) ornot. If the shutdown time>the predetermined time, then since thetemperature in the exhaust passage 3 and the tube wall thereof isconsidered to be substantially the same as the atmospheric temperature,the sensor temperature control means 18 sets the initial values Texg(0),Tga(0), Tgb(0), Tgc(0), Tgd(0), Twa(0), Twb(0), Twd(0), T_(O2)(0), andTht(0) to the detected value of the atmospheric temperature TA upon thestartup of the engine 1. If the shutdown time≦the predetermined time,then since the temperature in the exhaust passage 3 and the tube wallthereof is considered to be closer to the engine temperature TW (coolanttemperature) than to the atmospheric temperature, because of theafterheat of the engine 1 after the engine 1 has stopped its precedingoperation, the sensor temperature control means 18 sets the initialvalues Texg(0), Tga(0), Tgb(0), Tgc(0), Tgd(0), Twa(0), Twb(0), Twd(0),TO2(0), and Tht(0) to the detected value of the engine temperature TWupon the startup of the engine 1. These initial values are thus set to atemperature close to the actual temperature.

When the engine 1 starts to operate, the sensor temperature controlmeans 18 executes a main routine shown in FIG. 8 in a predeterminedcycle time. The period in which the main routine is executed is shorterthan the period dt of the processing sequence of the exhaust temperatureobserver 19 and the element temperature observer 20 and hence shorterthan the period dtc of the processing sequence of the target valuesetting means 21 and the heater controller 22.

The sensor temperature control means 18 acquires detected values of therotational speed NE and the intake pressure PB of the engine 1, theatmospheric temperature TA and the battery voltage VB in STEP1, and thendetermines the value of a countdown timer COPC for measuring the timedtc of one period of the processing sequence of the target value settingmeans 21 and the heater controller 22 in STEP2. The value of thecountdown timer COPC has been initialized to “0” at the time when theengine 1 starts to operate.

If COPC=0, then the sensor temperature control means 18 newly sets thevalue of the countdown timer COPC to a timer setting time TM1 whichcorresponds to the period dtc of the control processes of the targetvalue setting means 21 and the heater controller 22 in STEP3.Thereafter, the target value setting means 21 carries out a process ofsetting a target value R for the element temperature T_(O2) of the O₂sensor 8 in STEP4, and the heater controller 22 carries out a process ofcalculating a duty cycle DUT of The heater 13 in STEP5. If COPC≠0 inSTEP2, then the sensor temperature control means 18 counts down thevalue of the countdown timer COPC in STEP6, and skips the processing inSTEP4 and STEP5. Therefore, the processing in STEP4 and STEP5 is carriedout at the period dtc determined by the timer setting time TM1.

The processing in STEP4 and STEP5 is specifically carried out asfollows: First, the processing in STEP4 is carried out by the targetvalue setting means 21 as shown in FIG. 9.

The target value setting means 21 compares the value of a parameter TSHrepresentative of the time that has elapsed from the start of the engine1 with a predetermined value XTM (e.g., 15 seconds) in STEP4-1. IfTSH=XTM, i.e., if the engine 1 is in a state immediately after it hasstarted to operate, then the target value setting means 21 sets thetarget value R for the element temperature T_(O2) to a low temperature(e.g., 600° C.) in order to prevent damage to the active element 10 ofthe O₂ sensor 8 in STEP4-2. Specifically, the target value R that is setat this time is a target value R(n+Mr) after the target value predictingtime Mr from the present.

If TSH>XTM in STEP4-1, then the target value setting means 21 sets thetarget value R for the element temperature T_(O2) from the presentdetected value (acquired in STEP1 shown in FIG. 8) of the atmospherictemperature TA based on a predetermined table in STEP4-3. The targetvalue R that is set at this time is basically a predetermined value(800° C. in the present embodiment) equal to or higher than 750° C. ifthe atmospheric temperature TA is a normal temperature (e.g., TA≧0° C.).When the atmospheric temperature TA is low (e.g., TA<0° C.) as when theengine 1 is operating in a cold climate, if the target value R for theelement temperature T_(O2) is a high temperature of 800° C., thetemperature of the heater 13 is liable to be excessively high. In thepresent embodiment, when the temperature of the heater 13 becomesexcessively high, the heater 13 is forcibly de-energized by anoverheating prevention process (described later on) to prevent itselffrom a failure. In STEP4-3, according to the present embodiment, whenthe atmospheric temperature TA is low (e.g., TA<0° C.), the target valueR for the element temperature T_(O2) is set to a value slightly lowerthan the normal value (e.g., 750° C.≦R<800° C.).

Specifically, as with the target value R set in STEP4-2, the targetvalue R set in STEP4-3 is a target value R(n+Mr) after the target valuepredicting time Mr from the present.

After having set the target value R (=R(n+Mr)) in STEP4-2 or STEP4-3,the target value setting means 21 updates the values of Mr+1 buffersRBF(0), RBF(1), . . . , RBF(Mr) for storing target values R for thetarget value predicting time Mr in STEP4-4, STEP4-5. The processing inSTEP4 is now finished.

In STEP4-4, specifically, the Mr buffers RBF(j) (j=0, 1, . . . , Mr−1)are updated from the values of RBF(j) to the values of RBF(j+1), and thevalue held in the buffer RBF(0) so far is erased. In STEP4-5, the bufferRBF(Mr) is updated to the target value newly set in STEP4-2 or STEP4-3.The values of the buffers RBF(0), RBF(1), . . . , RBF(Mr) thus updatedcorrespond respectively to R(n), R(n+1), . . . , R(n+Mr) in the fourthterm of the equation (24). The values of the buffers RBF(0), RBF(1), . .. , RBF(Mr) have been initialized to a predetermined value (e.g., thetarget value set in STEP4-2) at the time the engine 1 has started tooperate.

The processing in STEP5 is carried out by the heater controller 22 asshown in FIG. 10. The heater controller 22 calculates an elementtemperature difference e(n)=T_(O2)(n)−RBF(0) between the presentestimated value T_(O2)(n) of the element temperature T_(O2) and thevalue of the buffer RBF(0) (=R(n)), i.e., the target value R set by thetarget value setting means 21 prior to the target value predicting timeMr in STEP5-1.

Then, the heater controller 22 determines the values of flags F/A, F/Bin STEP5-2. The flag F/A is set to “0” or “1” in a limiting process(described later on) for limiting the duty cycle DUT. The flag F/A whichis set to “1” means that the duty cycle DUT is forcibly limited to apredetermined upper or lower limit value, and the flag F/B which is setto “0” means that the duty cycle DUT is not limited to the predeterminedupper or lower limit value (the upper limit value>DUT>the lower limitvalue). The flag F/B is set to “1” when the heater 13 is forciblyde-energized by the overheating prevention process. The flags F/A, F/Bare initially set to “0”.

If F/A=F/B=0 in STEP5-2, then the heater controller 22 adds the presentvalue of Σe(j) in the first term of the equation (24) to the differencee(n) calculated in STEP5-1 in STEP5-3. In this manner, the differencee(n) is cumulatively added (integrated) in each cycle time dtc of theprocessing sequence of the heater controller 22. The initial value ofΣe(j) is “0”.

If F/A=1 or F/B=1 in STEP5-2, then since the present value of the dutycycle DUT is not a normal value, the heater controller 22 skips theprocessing in STEP5-2, but goes to STEP5-4, holding the present value ofΣe(j).

Then, the heater controller 22 calculates the equations (24), (25) usingthe present value (latest value) of the element temperature differencee(n) determined in STEP5-1 and the present accumulated value of Σe(j),thus calculating the present value DUT(n) of the control input DUT forthe heater 13 in STEP5-3. Specifically, the heater controller 22calculates the basic duty cycle SDUT(n) according to the equation (24)from the present value of the difference e(n) determined in STEP5-1, thepresent accumulated value Σe(j), the present estimated value Tht(n) ofthe heater temperature Tht, the present values (=R(n), R(n+1), . . . ,R(n+Mr)) of the buffers RBF(0), RBF(1), . . . , RBF(Mr), the presentestimated value Tgd(n) of the exhaust gas temperature Tgd (the exhaustgas temperature at the location of the O₂ sensor 8), and the values ofpredetermined coefficients Fs0, Fe0, Fx0, Fr(i) (i=0, 1, . . . , Mr),Fdt. The heater controller 22 then calculates the duty cycle DUT(n) bycorrecting the basic duty cycle SDUT(n) using the present value (thelatest value acquired in STEP1 shown in FIG. 8) of the battery voltageVB. The initial value Tht(0) of the estimated value of the heatertemperature Tht and the initial value Tgd(0) of the estimated value ofthe exhaust gas temperature Tgd, which are required for the firstprocessing of STEP5-4 (at a stage where the processing sequences of theexhaust temperature observer 19 and the element temperature observer 20are not executed) after the engine 1 has started to operate, are set tothe atmospheric temperature TA or the engine temperature TW when theengine 1 starts to operate (an engine startup). Those initial valuesTht(0), Tgd(0) are used in the calculation of the equation (24). Afterthe processing sequences of the exhaust temperature observer 19 and theelement temperature observer 20 are executed, the latest estimatedvalues of the estimated values that are determined in the processingsequences of the exhaust temperature observer 19 and the elementtemperature observer 20 are used in the calculation of the equation(24).

Then, the heater controller 22 carries out a limiting process forlimiting the duty cycle DUT(n) calculated in STEP5-4 in STEP5-5 throughSTEP5-11. Specifically, the heater controller 22 determines whether theduty cycle DUT(n) is smaller than a predetermined lower limit value(e.g., “0%”) or not in STEP5-5. If DUT(n)<the lower limit value, thenthe heater controller 22 forcibly sets the value of DUT(n) to the lowerlimit value in STEP5-6. Thereafter, the value of the flag F/A (the flagused in STEP5-2) is set to “1” in STEP5-7.

If DUT(n)≧the lower limit value, then the heater controller 22determines whether the duty cycle DUT(n) is greater than a predeterminedupper limit value (e.g., 100%) or not in STEP5-8. If DUT(n)>the upperlimit value, then the heater controller 22 forcibly sets the value ofDUT(n) to the upper limit value in STEP5-9. Thereafter, the value of theflag F/A is set to “1” in STEP5-10. If the lower limit value≦DUT(n)≦theupper limit value, then the heater controller 22 holds the value ofDUT(n), and sets the flag F/A to “0” in STEP5-11. The processing inSTEP5 is now finished.

Control then returns to the main routine shown in FIG. 8. The sensortemperature control means 18 carries out the processing in STEP7 throughSTEP13. The processing in STEP7 through STEP13 represents a process ofpreventing the heater 13 from being overheated. In STEP7, the sensortemperature control means 18 determines whether or not the presentestimated value (latest value) of the heater temperature Tht is equal toor higher than a predetermined upper limit value THTLMT (e.g., 930° C.).In the present embodiment, if Tht≧THTLMT, the sensor temperature controlmeans 18 forcibly de-energizes the heater 13 to prevent the heater 13from being damaged. However, the estimated value of Tht may temporarilyrise to a value equal to or higher than the upper limit value THTLMT dueto a disturbance or the like. According to the present embodiment,therefore, the sensor temperature control means 18 forcibly de-energizesthe heater 13 if the state in which Tht=THTLMT has continued for apredetermined time (e.g., 3 seconds, hereinafter referred to as “heaterOFF delay time”).

If Tht<THTLMT in STEP7, then the sensor temperature control means 18sets the value of a countdown timer TMHTOFF for measuring the heater OFFdelay time to a predetermined value TM2 corresponding to the heater OFFdelay time in STEP8. Since the sensor temperature control means 18 doesnot forcibly de-energize the heater 13 at this time, the sensortemperature control means 18 sets the value of the flag-F/B (the flagused in STEP5-2 shown in FIG. 10) to “0” in STEP9.

If Tht=THTLMT in STEP7, then the sensor temperature control means 18counts down the value of the countdown timer TMHTOFF by “1” in STEP10.Then, the sensor temperature control means 18 determines whether thevalue of the countdown timer TMHTOFF is “0” or not, i.e., whether theheater OFF delay time TM2 has elapsed with Tht=THTLMT or not in STEP11.

If TMHTOFF≠0, then the sensor temperature control means 18 sets the flagF/B to “0” in STEP9. If TMHTOFF=0, then the sensor temperature controlmeans 18 forcibly sets the present value of the duty cycle DUT to “0” inSTEP12, and then sets the value of the flag F/B to “1” in STEP13.

When the flag F/B is set to “0” in STEP9, the sensor temperature controlmeans 18 applies a pulsed voltage to the heater energization circuitaccording to the present value of the duty cycle DUT (the latest valuecalculated in STEP5), energizing the heater 13 with the electric energydepending on the duty cycle DUT. When the value of the flag F/B is setto “1” in STEP12, the sensor temperature control means 18 does not applya pulsed voltage to the heater energization circuit, thus de-energizingthe heater 13.

After having thus executed the processing in STEP7 through STEP13, i.e.,the process of preventing the heater 13 from being overheated, thesensor temperature control means 18 determines the value of a countdowntimer COBS for measuring the time dt of one period of the processingsequences of the exhaust temperature observer 19 and the elementtemperature observer 20 in STEP14. The value of the countdown timer COBSis initially set to “0” when the engine 1 has started to operate.

If COBS=0, then the sensor temperature control means 18 newly sets thevalue of COBS to a timer setting time TM3 (shorter than TM1 in STEP3)which corresponds to the period dt of the processing sequences of theexhaust temperature observer 19 and the element temperature observer 20in STEP15. Then, the exhaust temperature observer 19 carries out aprocess of estimating the exhaust gas temperature Tgd (the exhaust gastemperature in the vicinity of the location of the O₂ sensor 8), and theelement temperature observer 20 carries out a process of estimating theelement temperature T_(O2) (including a process of estimating the heatertemperature Tht) in STEP16. If COBS≠0 in STEP14, the sensor temperaturecontrol means 18 counts down the value of COBC in STEP17, skipping theprocessing in STEP15 and STEP16. The processing in STEP16 is thereforecarried out at a period dt which is determined by the timer setting timeTM3. The main routine shown in FIG. 8 is now finished.

The processing in STEP16 is specifically carried out as shown in FIG.11. The exhaust temperature observer 19 successively carries out theprocessing in STEP16-1 through STEP16-6 to determine an estimated valueof the exhaust gas temperature Tgd in the vicinity of the location ofthe O₂ sensor 8. In STEP16-1, the exhaust temperature observer 19determines a gas speed parameter Vg according to the equation (7) usingthe present detected values (the latest values acquired in STEP1) of therotational speed NE and the intake pressure PB of the engine 1. The gasspeed parameter Vg is forcibly set to Vg=1 if the result calculated bythe equation (7) exceeds “1” due to an excessive rotational speed of theengine 1.

Then, the exhaust temperature observer 19 calculates an estimated valueof the exhaust gas temperature Texg at the exhaust port 2 of the engine1 according to the equation (1) in STEP16-2. Specifically, the exhausttemperature observer 19 determines a basic exhaust gas temperatureTMAP(NE,PB) from the present detected values of the rotational speed NEand the intake pressure PB of the engine 1 based on the predeterminedmap, and thereafter calculates the right side of the equation (1) usingthe basic exhaust gas temperature TMAP(NE,PB), the present estimatedvalue Texg(k−1) (determined in STEP16-2 in the preceding cycle time) ofthe exhaust gas temperature Texg, and the value of a predeterminedcoefficient Ktex, thus calculating a new estimated value Texg(k) of theexhaust gas temperature Texg. In the present embodiment, while theengine 1 is idling and also while the supply of fuel to the engine 1 isbeing cut off, the basic exhaust gas temperature TMAP used in thecalculation of the equation (1) is set to predetermined valuescorresponding to the respective engine operating states. When the engine1 starts to operate (upon an engine startup), the atmospherictemperature TA or the engine temperature TW detected at this time is setas an initial value Texg(0) of the estimated value of the exhaust gastemperature Texg. When the equation (1) is calculated for the first timeafter the engine 1 has started to operate, the initial value Texg(0) isused as the value of Texg(k−1).

Then, the exhaust temperature observer 19 calculates an estimated valueof the exhaust gas temperature Tga and an estimated value of the exhaustpipe temperature Twa in the partial exhaust passageway 3 a according tothe respective equations (5-1), (5-2) in STEP16-2. Specifically, theexhaust temperature observer 19 determines a new estimated valueTga(k+1) of the exhaust gas temperature Tga by calculating the rightside of the equation (5-1) using the present estimated value Tga(k)(determined in STEP16-3 in the preceding cycle time) of the exhaust gastemperature Tga, the present estimated value (determined in STEP16-3 inthe preceding cycle time) of the exhaust pipe temperature Twa, thepresent estimated value of the exhaust gas temperature Texg previouslycalculated in STEP16-2, the present value of the gas speed parameter Vgcalculated in STEP16-1, the value of the predetermined model coefficientAa, and the value of the period dt of the processing sequence of theexhaust temperature observer 19.

The exhaust temperature observer 19 calculates a new estimated valueTwa(k+1) of the exhaust pipe temperature Twa by calculating the rightside of the equation (5-2) using the present estimated value Tga(k)(determined in STEP16-3 in the preceding cycle time) of the exhaust gastemperature Tga, the present estimated value (determined in STEP16-3 inthe preceding cycle time) of the exhaust pipe temperature Twa, thevalues of the predetermined model coefficients Ba, Ca, and the value ofthe period dt of the processing sequence of the exhaust temperatureobserver 19.

When the engine 1 starts to operate (upon an engine startup), theatmospheric temperature TA or the engine temperature TW detected at thistime is set as initial values Tga(0), Twa(0) of the estimated values ofthe exhaust gas temperature Tga and the exhaust pipe temperature Twa.When the equations (5-1), (5-2) are calculated for the first time afterthe engine 1 has started to operate, these initial values Tga(0), Twa(0)are used as the respective values of Tga(k−1), Twa(k−1).

Then, the exhaust temperature observer 19 calculates an estimated valueof the exhaust gas temperature Tgb and an estimated value of the exhaustpipe temperature Twb in the partial exhaust passageway 3 b according tothe respective equations (6-1), (6-2) in STEP16-4. Specifically, theexhaust temperature observer 19 determines a new estimated valueTgb(k+1) of the exhaust gas temperature Tgb by calculating the rightside of the equation (6-1) using the present estimated value Tgb(k) (thelatest value determined in STEP16-4 in the preceding cycle time) of theexhaust gas temperature Tgb, the present estimated value (the latestvalue determined in STEP16-4 in the preceding cycle time) of the exhaustpipe temperature Twb, the present estimated value of the exhaust gastemperature Tga previously calculated in STEP16-3, the present value ofthe gas speed parameter Vg calculated in STEP16-1, the value of thepredetermined model coefficient Ab, and the value of the period dt ofthe processing sequence of the exhaust temperature observer 19.

The exhaust temperature observer 19 calculates a new estimated valueTwb(k+1) of the exhaust pipe temperature Twb by calculating the rightside of the equation (6-2) using the present estimated value Tgb(k) (thelatest value determined in STEP16-4 in the preceding cycle time) of theexhaust gas temperature Tgb, the present estimated value (the latestvalue determined in STEP16-4 in the preceding cycle time) of the exhaustpipe temperature Twb, the values of the predetermined model coefficientsBb, Cb, and the value of the period dt of the processing sequence of theexhaust temperature observer 19.

When the engine 1 starts to operate, the atmospheric temperature TA orthe engine temperature TW detected at this time is set as initial valuesTgb(0), Twb(0) of the estimated values of the exhaust gas temperatureTgb and the exhaust pipe temperature Twb. When the equations (6-1),(6-2) are calculated for the first time after the engine 1 has startedto operate, these initial values Tgb(0), Twb(0) are used as therespective values of Tgb(k−1), Twb(k−1).

Then, the exhaust temperature observer 19 calculates an estimated valueof the exhaust gas temperature Tgc and an estimated value of thecatalyst temperature Twc in the partial exhaust passageway 3 c accordingto the respective equations (8-1), (8-2) in STEP16-5. Specifically, theexhaust temperature observer 19 determines a new estimated valueTgc(k+1) of the exhaust gas temperature Tgc by calculating the rightside of the equation (8-1) using the present estimated value Tgc(k) (thelatest value determined in STEP16-5 in the preceding cycle time) of theexhaust gas temperature Tgc, the present estimated value (the latestvalue determined in STEP16-5 in the preceding cycle time) of thecatalyst temperature Twc, the present estimated value of the exhaust gastemperature Tgb previously calculated in STEP16-4, the present value ofthe gas speed parameter Vg calculated in STEP16-1, the value of thepredetermined model coefficient Ac, and the value of the period dt ofthe processing sequence of the exhaust temperature observer 19.

The exhaust temperature observer 19 calculates a new estimated valueTwc(k+1) of the catalyst temperature Twc by calculating the right sideof the equation (8-2) using the present estimated value Tgc(k) (thelatest value determined in STEP16-5 in the preceding cycle time) of theexhaust gas temperature Tgc, the present estimated value (the latestvalue determined in STEP16-5 in the preceding cycle time) of thecatalyst temperature Twc, the present value of the gas speed parameterVg calculated in STEP16-1, the values of the predetermined modelcoefficients Bc, Cc, Dc, and the value of the period dt of theprocessing sequence of the exhaust temperature observer 19.

When the engine 1 starts to operate, the atmospheric temperature TA orthe engine temperature TW detected at this time is set as initial valuesTgc(0), Twc(0) of the estimated values of the exhaust gas temperatureTgc and the catalyst temperature Twc. When the equations (8-1), (8-2)are calculated for the first time after the engine 1 has started tooperate, these initial values Tgc(0), Twc(0) are used as the respectivevalues of Tgc(k−1), Twc(k−1).

Then, the exhaust temperature observer 19 calculates an estimated valueof the exhaust gas temperature Tgd and an estimated value of the exhaustpipe temperature Twd in the partial exhaust passageway 3 d (near thelocation of the O₂ sensor 8) according to the respective equations(9-1), (9-2) in STEP16-6. Specifically, the exhaust temperature observer19 determines a new estimated value Tgd(k+1) of the exhaust gastemperature Tgd by calculating the right side of the equation (9-1)using the present estimated value Tgd(k) (the latest value determined inSTEP16-6 in the preceding cycle time) of the exhaust gas temperatureTgd, the present estimated value (the latest value determined inSTEP16-6 in the preceding cycle time) of the exhaust pipe temperatureTwd, the present estimated value of the exhaust gas temperature Tgcpreviously calculated in STEP16-5, the present value of the gas speedparameter Vg calculated in STEP16-1, the value of the predeterminedmodel coefficient Ad, and the value of the period dt of the processingsequence of the exhaust temperature observer 19.

The exhaust temperature observer 19 calculates a new estimated valueTwd(k+1) of the exhaust pipe temperature Twd by calculating the rightside of the equation (9-2) using the present estimated value Tgd(k) (thelatest value determined in STEP16-6 in the preceding cycle time) of theexhaust gas temperature Tgd, the present estimated value (the latestvalue determined in STEP16-6 in the preceding cycle time) of the exhaustpipe temperature Twd, the values of the predetermined model coefficientsBd, Cd, and the value of the period dt of the processing sequence of theexhaust temperature observer 19.

When the engine 1 starts to operate, the atmospheric temperature TA orthe engine temperature TW detected at this time is set as initial valuesTgd(0), Twd(0) of the estimated values of the exhaust gas temperatureTgd and the exhaust pipe temperature Twd. When the equations (9-1),(9-2) are calculated for the first time after the engine 1 has startedto operate, these initial values Tgd(0), Twd(0) are used as therespective values of Tgd(k−1), Twd(k−1).

Then, the element temperature observer 20 executes the processing inSTEP16-7 to determine estimated values of the element temperature T_(O2)of the O₂ sensor 8 and the heater temperature Tht according to theequations (10-1), (10-2). Specifically, the element temperature observer20 determines a new estimated value T_(O2)(k+1) of the elementtemperature T_(O2) by calculating the right side of the equation (10-1)using the present estimated value T_(O2)(k) (the latest value determinedin STEP16-7 in the preceding cycle time) of the element temperatureT_(O2), the present estimated value Tht(k) (the latest value determinedin STEP16-7 in the preceding cycle time) of the heater temperature Tht,the present estimated value Tgd(k) of the exhaust gas temperature Tgdpreviously calculated in STEP16-6, the present value TA(k) (the latestvalue acquired in STEP1 shown in FIG. 8) of the detected value of theatmospheric temperature TA as the temperature TA′ of the air in theactive element 10, the values of the predetermined model coefficientsAx, Bx, and the value of the period dt (=the period of the processingsequence of the exhaust temperature observer 19) of the processingsequence of the element temperature observer 20.

Then, the element temperature observer 20 determines a new estimatedvalue Tht(k+1) of the heater temperature Tht by calculating the rightside of the equation (10-2) using the present estimated value T_(O2)(k)(the latest value determined in STEP16-7 in the preceding cycle time) ofthe element temperature T_(O2), the present estimated value Tht(k) (thelatest value determined in STEP16-7 in the preceding cycle time) of theheater temperature Tht, the present value TA(k) (the latest valueacquired in STEP1 shown in FIG. 8) of the detected value of theatmospheric temperature TA as the temperature TA′ of the air in theactive element 10, the present value DUT(k) of the duty cycle DUT, thevalues of the predetermined model coefficients Cx, Dx, and the value ofthe period dt of the processing sequence of the element temperatureobserver 20.

When the engine 1 starts to operate, the atmospheric temperature TA orthe engine temperature TW detected at this time is set as initial valuesT_(O2)(0), Tht(0) of the estimated values of the element temperatureT_(O2) and the heater temperature Tht. When the equations (10-1), (10-2)are calculated for the first time after the engine 1 has started tooperate, these initial values T_(O2)(0), Tht(0) are used as therespective values of T_(O2)(k−1), Tht(k−1). The duty cycle DUT(k) usedin the equation (10-2) is basically of the latest value determined bythe heater controller 22 in STEP5. However, if the value of the dutycycle DUT is limited in STEP12 to “0”, i.e., to de-energize the heater13, then the limited value of the duty cycle DUT is used in the equation(10-2).

The above processing sequence of the sensor temperature control means 18controls the electric energy supplied to the heater 13 of the O₂ sensor8 in order to keep the element temperature T_(O2) of the O₂ sensor 8 atthe target value R. Except immediately after the engine 1 has started tooperate and when the atmospheric temperature TA is considerably low, thetarget value R is normally set to 800° C. As a result, the outputcharacteristics of the O₂ sensor 8 can be maintained stably as thecharacteristics suitable for controlling the air-fuel ratio of theengine 1, i.e., for controlling the air-fuel ratio thereof for thecatalytic converter 4 to perform a better exhaust purifying capability,and the air-fuel ratio of the engine 1 can well be controlled to allowthe catalytic converter 4 to perform a better exhaust purifyingcapability.

According to the present embodiment, the duty cycle DUT as a controlinput for the heater 13 includes a control input component (the firstand second terms of the equation (24)) depending on the difference(element temperature difference) e between the estimated value of theelement temperature T_(O2) and the target value R, and also includes acontrol input component (the third term of the equation (24) dependingon the estimated value of the heater temperature Tht. Therefore, whenthe element temperature T_(O2) varies with respect to the target valueR, it is possible to converge the element temperature T_(O2) smoothly tothe target value R while preventing the duty cycle DUT of the heater 13from being excessively changed. According to the present embodiment,since ΔTht of the heater temperature Tht is a state quantity of themodel to be controlled, the control input component depending on theestimated value of the heater temperature Tht is meant to be a feedbackcomponent.

According to the present embodiment, furthermore, the duty cycle DUTalso includes a control input component depending on the estimated valueof the exhaust gas temperature Tgd which acts as a disturbant factor forvarying the element temperature T_(O2), i.e., the optimum disturbanceF/F component Uopfd. The coefficient Fdt relative to the optimumdisturbance F/F component Uopfd is determined according to a predictivecontrol algorithm on the assumption that the present exhaust gastemperature will continue until after the exhaust gas temperaturepredicting time Md. Consequently, it is possible to control the elementtemperature T_(O2) at the target value R while suppressing variations ofthe element temperature T_(O2) due to variations of the exhaust gastemperature Tgd. In particular, as the coefficient Fdt relative to theoptimum disturbance F/F component Uopfd is determined according to apredictive control algorithm, any variations of the element temperatureT_(O2) due to variations of the exhaust gas temperature Tgd can beminimized. As a result, the stability of the process of controlling theelement temperature T_(O2) at the target value R is effectivelyincreased, and hence the stability of the output characteristics of theO₂ sensor 8 is also effectively increased.

Furthermore, the control input DUT includes the control input componentdepending on the target value R for the element temperature T_(O2),i.e., the optimum target value F/F component Uopfr. In addition, theoptimum target value F/F component Uopfr is turned into a control inputcomponent depending on the target value R from the present until afterthe target value predicting time Mr by the predictive control algorithm.Therefore, when the target value R changes from a low temperature (600°C.) immediately after the engine 1 has started to operate to a normalhigh temperature (750° C. through 800° C.) in particular, the controlinput DUT is prevented from becoming temporarily large excessively, andthe element temperature T_(O2) is prevented from overshooting withrespect to the target value R. The stability of the outputcharacteristics of the O₂ sensor 8 is also effectively increased.

FIGS. 12 and 13 show the results of simulations of the presentembodiment. FIG. 12 illustrates how the heater temperature Tht, theelement temperature T_(O2), and the duty cycle DUT change when thetarget value R changes from a low-temperature target value (600° C.) toa high-temperature target value (800° C.) at a time t1 while a vehiclecarrying the engine 1 is running at a constant speed (in a steadyoperating state of the engine 1). In FIG. 12, the solid-line curves e,g, i indicate how the heater temperature Tht, the element temperatureT_(O2), and the duty cycle DUT, respectively, change when the heater 13is controlled according to the present embodiment. The broken-linecurves f, h, j indicate how the heater temperature Tht, the elementtemperature T_(O2), and the duty cycle DUT, respectively, change in acomparative example. According to the comparative example, the dutycycle DUT is determined using an equation similar to the equation (24)except that the optimum target value F/F component Uopfr (the fourthterm of the equation (24)) depending on the target value R is removedfrom the equation (24).

It can be seen from FIG. 24 that according to the present embodiment,since the duty cycle DUT includes the optimum target value F/F componentUopfr depending on the target value R, when the target value R switches,the duty cycle DUT is prevented from becoming excessively large, andtransient overshooting of the heater temperature Tht and the elementtemperature T_(O2) is reduced.

FIG. 13 shows how the element temperature T_(O2) changes while a vehiclecarrying the engine 1 is running at varying vehicle speeds as indicatedby a lower curve in FIG. 13 (in various changing operating states of theengine 1). In FIG. 13, the solid-line curve p indicates how the elementtemperature T_(O2) changes when the heater 13 is controlled according tothe present embodiment. The broken-line curve q indicates how theelement temperature T_(O2) changes in a comparative example. Accordingto the comparative example, the duty cycle DUT is determined using anequation similar to the equation (24) except that the optimumdisturbance F/F component Uopfd (the fifth term of the equation (24))depending on the exhaust gas temperature Tgd is removed from theequation (24).

It can be seen from FIG. 13 that according to the present embodiment,since the duty cycle DUT includes the optimum disturbance F/F componentUopfd depending on the exhaust gas temperature Tgd, a range ofvariations of the element temperature T_(O2) due to changes in theexhaust gas temperature Tgd is reduced.

A second embodiment of the present invention will be described belowwith reference to FIG. 14. The second embodiment is partly different inarrangement or function from the first embodiment described above, andthose structural or functional parts of the second embodiment which areidentical to those of the first embodiment are denoted by identicalreference characters, and will not be described in detail below.

According to the present embodiment, as shown in the block diagram ofFIG. 14, the sensor temperature control means 18 of the control unit 16shown in FIG. 1 comprises, as functional means, an exhaust temperatureobserver 19, an element temperature observer 20, a target value settingmeans 31, and a heater controller 32. The exhaust temperature observer19 and the element temperature observer 20 are identical to those of thefirst embodiment. In the present embodiment, the target value settingmeans 31 and the heater controller 32 have their processing periodsidentical to the processing periods of the target value setting means 21and the heater controller 22 according the first embodiment.

The target value setting means 31 serves to set a target value R′ forthe heater temperature Tht of the O₂ sensor 8. According to theinventors' knowledge, the heater temperature Tht is relatively highlycorrelated to the element temperature T_(O2) and tends to be higher thanthe element temperature T_(O2) by a constant temperature in a steadystate. According to the present embodiment, the target value settingmeans 31 sets, as the target value R′ for the heater temperature Tht, avalue R+DR which is higher than the target value R for the elementtemperature T_(O2) that is set as described in the first embodiment (thetarget value R set by the processing sequence shown in FIG. 9), by apredetermined value DR (e.g., 100° C.). As with the first embodiment,the target value R′ that is set by the target value setting means 31 ineach cycle time of its processing sequence is a target value after thetarget value predicting time Mr, and the target value R′ in the periodof the target value predicting time Mr is sequentially updated andstored.

The heater controller 32 sequentially generates the duty cycle DUT as acontrol input in order to keep the heater temperature Tht at the targetvalue R′. In the present embodiment, as with the first embodiment, theheater controller 32 calculates a basic duty cycle SDUT according to anoptimum predictive control algorithm, and corrects the basic duty cycleSDUT depending on the battery voltage VB according to the equation (25)thereby to generate the duty cycle DUT.

More specifically, according to the present embodiment, attention ispaid to the difference e′ between the heater temperature Tht and atarget value R′ therefor, a change Δe′ per given time in the differencee′ (corresponding to a rate of change of the difference e′), and achange ΔT_(O2) per given time in the element temperature T_(O2)(corresponding to a rate of change of the element temperature T_(O2)),and a model equation for an object to be controlled by the heatercontroller 32 is introduced using the above differences and changes asstate quantities relative to the object to be controlled by the heatercontroller 32.

If the difference e′ (hereinafter referred to as “heater temperaturedifference e′”) is defined as e′(n)=Tht(n)−R′(n), then the modelequation is given as the following equation (26) based on the aboveequations (11-1), (11-2) according to the same idea as with the firstembodiment:

$\begin{matrix}{{{X\; 1\left( {n + 1} \right)} = {{{\Phi^{\prime} \cdot X}\; 1(n){G^{\prime} \cdot \Delta}\;{{SDUT}(n)}} + {{{Gd}^{\prime} \cdot \Delta}\;{{Tgd}(n)}} + {{{Gr}^{\prime} \cdot R}\; 1\left( {n + 1} \right)}}}{where}{{{X\; 1(n)} = \left( {{e^{\prime}(n)},{\Delta\;{e^{\prime}(n)}},{\Delta\;{T_{O\mspace{11mu} 2}(n)}}} \right)^{T}},{{R\; 1\left( {n + 1} \right)} = \left( {{\Delta\;{R^{\prime}\left( {n + 1} \right)}},{\Delta\;{R^{\prime}(n)}}} \right)^{T}},{G^{\prime} = \left( {0,{{Dx} \cdot {dtc}},0} \right)^{T}},{{Gd}^{\prime} = \left( {0,0,{{Ax} \cdot {dtc}}} \right)^{T}},{\Phi^{\prime} = \begin{bmatrix}1 & 1 & 0 \\0 & {1 - {{Cx} \cdot {dtc}} - {{Fx} \cdot {dtc}}} & {{Cx} \cdot {dtc}} \\0 & {{Bx} \cdot {dtc}} & {1 - {{Ax} \cdot {dtc}} - {{Bx} \cdot {dtc}} - {{Ex} \cdot {dtc}}}\end{bmatrix}}}{{Gr}^{\prime} = \begin{bmatrix}0 & 0 \\{- 1} & {1 - {{Cx} \cdot {dtc}} - {{Fx} \cdot {dtc}}} \\0 & {{Bx} \cdot {dtc}}\end{bmatrix}}} & (26)\end{matrix}$

In the present embodiment, the basic control input SDUT (a control inputat the time the battery voltage VB is equal to the reference value NVB)to be determined by the heater controller 22 is given by the equation(28) shown below as having integrated ΔSDUT (a control input on themodel according to the equation (26)) which minimizes an evaluatingfunction J1 according to the following equation (27):

$\begin{matrix}{{{J\; 1} = {\sum\limits_{n = {M + 1}}^{\infty}\;\left\lbrack {{X\; 1^{T}{(n) \cdot Q}\;{0 \cdot X}\; 1(n)} + {\Delta\;{{{SDUT}^{T}(n)} \cdot H}\;{0 \cdot \Delta}\;{{SDUT}(n)}}} \right\rbrack}}{{{where}\mspace{14mu} M} = {\max\left( {{Mr},{Md}} \right)}}} & (27) \\{{{SDUT}(n)} = {{{Fs}\; 1} + {\sum\limits_{j = 1}^{n}\;{e^{\prime}(j)}} + {{Fe}\;{1 \cdot {e^{\prime}(n)}}} + {{Fx}\;{1 \cdot {T_{O\; 2}(n)}}} + {\sum\limits_{i = 0}^{Mr}\;\left\lbrack {{{Fr}^{\prime}(i)} \cdot {R^{\prime}\left( {n + i} \right)}} \right\rbrack} + {{Fdt}^{\prime} \cdot {{Tgd}(n)}}}} & (28)\end{matrix}$

The coefficients Fs1, Fe1, Fx1 in the first through third terms, thecoefficient Fr1(i) (i=0, 1, . . . , Mr) in the fourth term, and thecoefficient Fdt′ in the fifth term on the right side of the equation(28) are coefficients given respectively by the following equations(29-1) through (29-3):

$\begin{matrix}\begin{matrix}{{F\; 1} \equiv} & {\left( {{{Fs}\; 1},\;{{Fe}\; 1},\;{{Fx}\; 1}} \right)} \\{=} & {{- \left\lbrack {{H\; 0} + {G^{\prime\; T} \cdot P^{\prime} \cdot G^{\prime}}} \right\rbrack^{- 1}} \cdot G^{\prime\; T} \cdot P^{\prime} \cdot \Phi^{\prime}}\end{matrix} & \left( {29\text{-}1} \right) \\{{{Fr}^{\prime}(i)} = \left\lbrack \begin{matrix}{{Fr}\; 12(1)} & {{:i} = 0} \\{{{Fr}\; 11(i)} + {{Fr}\; 12\left( {i + 1} \right)}} & {{{:i} = 1},2,\cdots,{{Mr} - 1}} \\{{Fr}\; 11({Mr})} & {{:i} = {Mr}}\end{matrix} \right.} & \left( {29\text{-}2} \right) \\{{{Fdt}^{\prime} = {\sum\limits_{i = 0}^{Md}\;\left\{ {{- \left\lbrack {{H\; 0} + {G^{\prime\; T} \cdot P^{\prime} \cdot G^{\prime}}} \right\rbrack^{- 1}} \cdot G^{\prime\; T} \cdot \left( \zeta^{T} \right)^{i} \cdot P^{\prime} \cdot {Gd}^{\prime}} \right\}}}{{{where}\mspace{14mu} P^{\prime}} = {{Q\; 0} + {\Phi^{\prime\; T} \cdot P^{\prime} \cdot \Phi^{\prime}} - {\Phi^{\prime} \cdot P^{\prime} \cdot G^{\prime} \cdot \left\lbrack {{H\; 0} + {G^{\prime\; T} \cdot P^{\prime} \cdot G^{\prime}}} \right\rbrack^{- 1} \cdot G^{\prime\; T} \cdot P^{\prime} \cdot \Phi^{\prime}}}}{\zeta^{\prime} = {\Phi^{\prime} + {{G^{\prime} \cdot F}\; 1}}}{\left( {{{Fr}\; 11(i)},{{Fr}\; 12(i)}} \right) = {{- \left\lbrack {{H\; 0} + {G^{\prime\; T} \cdot P^{\prime} \cdot G^{\prime}}} \right\rbrack^{- 1}} \cdot G^{\prime\; T} \cdot \left( \zeta^{\prime\; T} \right)^{i - 1} \cdot P^{\prime} \cdot {{Gr}^{\prime}\left( {{i = 1},2,\cdots,{Mr}} \right)}}}} & \left( {29\text{-}3} \right)\end{matrix}$

In the present embodiment, the weighted matrixes Q0, H0 with respect tothe evaluating function J1, the target value predicting time Mr, and theexhaust gas temperature predicting time Md are identical to those in thefirst embodiment. However, they may be set to values different fromthose in the first embodiment. The coefficients Fs1, Fe1, Fx1, Fr′(i),Fdt′ in the equation (28) may not necessarily be of the values accordingto the defining equations (29-1) through (29-3), but may be of valuesadjusted by way of simulation or experimentation. Furthermore, thecoefficients Fs1, Fe1, Fx1, Fr′(i), Fdt′ may be changed depending on theelement temperature, the heater temperature, etc. In the presentembodiment, as with the first embodiment, the exhaust gas temperatureTgd is maintained at the present value in the future until after Mdsteps. However, if Tgd at each time in the future can be detected orestimated, then the control input DUT may be determined using thosevalues (in this case, Fdt′ is a vector).

The above equation (28) is a formula for sequentially calculating abasic control input SDUT(n) with which the heater controller 32 controlsthe heater 13 in the present embodiment. Specifically, the heatercontroller 32 sequentially calculates the basic control input SDUT(n) ineach cycle time (period) of the control processing of the heatercontroller 32 according to the equation (28), and corrects the basiccontrol input SDUT(n) depending on the battery voltage VB according tothe equation (25), thereby determining the duty cycle DUT(n). The termson the right side of the equation (28) have the same meanings as thosein the first embodiment. Specifically, the first through third terms(the term including Σe′ (j) through the term including T_(O2)(n)) on theright side represent a control input component (a feedback componentbased on an optimum control algorithm) depending on the heatertemperature difference e′ and the element temperature T_(O2). Morespecifically, the first and second terms represent a control inputcomponent depending on the heater temperature difference e′, and thethird term represents a control input component depending on the elementtemperature T_(O2). The fourth term (the term of ΣFr′(i)·R′(n+1)) on theright side of the equation (28) and the fifth term (the term includingTgd(n)) on the right side thereof represent control input components(feed-forward components based on a predictive control algorithm)depending on the target value R and the exhaust gas temperature Tgd,respectively.

As the element temperature T_(O2) and the exhaust gas temperature Tgdwhich are required to determine the basic control input SDUT(n)according to the equation (28), there are employed, respectively, thelatest value of the estimated value of the element temperature T_(O2)determined by the element temperature observer 20 and the latest valueof the estimated value of the exhaust gas temperature Tgd determined bythe exhaust temperature observer 19.

The heater temperature difference e′ required for the calculationaccording to the equation (28) is calculated from the latest value ofthe estimated value of the heater temperature Tht determined by theelement temperature observer 20 and the target value R′ that has beenset in a cycle time before the target value predicting time Mr by thetarget value setting means 31.

The other processing details than those described above are identical tothose according to the first embodiment. In the present embodiment, theelectric power supplied to the heater 13 of the O₂ sensor 8 iscontrolled in order to maintain the heater temperature Tht of the O₂sensor 8 at the target value R′. In this case, except immediately afterthe engine 1 starts to operate or when the atmospheric temperature TA isconsiderably low (TA<0° C.), the target value R′ is usually set to atemperature (900° C.) which is higher than a preferred targettemperature of 800° C. for the active element 10 by a predeterminedvalue DR (100° C. in the present embodiment). As a result, thetemperature T_(O2) of the active element 10 of the O₂ sensor 8 isindirectly controlled substantially at the temperature of 800° C.Therefore, as with the first embodiment, the output characteristics ofthe O₂ sensor 8 can stably be kept as characteristics suitable forcontrolling the air-fuel ratio of the engine 1 (for controlling theair-fuel ratio to keep a good purifying capability of the catalyticconverter 4), and hence the air-fuel ratio is controlled well toreliably keep a good purifying capability of the catalytic converter 4.During a predetermined period of time immediately after the engine 1 hasstarted to operate, the target temperature R′ for the heater 13 is setto a temperature (700° C.) which is higher than a low temperature (600°C.) as the target temperature R for the active element 10 than thepredetermined value DR, for thereby preventing the active element 10from being damaged by stresses due to abrupt heating. If the atmospherictemperature T_(A) is low (TA<0° C.), then inasmuch as the target value Rfor the active element 10 is set to a value in the range of 750°C.≦R<800° C., the target value R′ for the heater 13 is set to a value inthe range of 850° C.≦R′<900° C. to prevent the heater 13 from beingoverheated.

According to the present embodiment, the duty cycle DUT as a controlinput to the heater 13 includes the control input component (the firstterm (the term including Σe′(j)) and the second term (the term includinge′(n)) of the equation (28)) depending on the difference between theestimated value of the heater temperature Tht an the target temperatureR′, and the control input component (the third term of the equation(28)) depending on the estimated value of the element temperatureT_(O2), as feedback components. In addition, according to the presentembodiment, a predictive control algorithm is also applied, and the dutycycle DUT includes the control input component (a feed-forward componentof the fifth term on the right side of the equation (28)) depending onthe exhaust gas temperature Tgd, and the control input component (afeed-forward component of the fourth term on the right side of theequation (28)) depending on the target value R′. As a result, thepresent embodiment provides the same operation and advantages as withthe first embodiment. Thus, the heater temperature Tht can reliably becontrolled stably at the desired target value R′, and the elementtemperature T_(O2) can be controlled stably at a desired temperature.

A third embodiment of the present invention will be described below. Thepresent embodiment is different from the first embodiment only as to theprocessing of the heater controller (specifically, the processing inSTEP5-4 shown in FIG. 10), and those structural or functional parts ofthe third embodiment which are identical to those of the firstembodiment are denoted by identical figures and reference characters,and will not be described in detail below.

According to the present embodiment, the heater controller 22sequentially generates DUT as a control input according to an optimumpredictive control algorithm, using the difference e (elementtemperature difference e) between the element temperature T_(O2) and thetarget value R, the element temperature T_(O2), and the heatertemperature Tht as state quantities to be controlled. An algorithm ofthe processing sequence of the heater controller 22 is constructed asfollows: The equations (11-1), (11-2) are brought together, providingthe following equations (30-1), (30-2):

$\begin{matrix}{{{Xz}\left( {k + 1} \right)} = {{{Az} \cdot {{Xz}(k)}} + {{{Bz} \cdot \Delta}\;{{SDUT}(k)}} + {{{Ez} \cdot \Delta}\;{{Tgd}(k)}}}} & \left( {30\text{-}1} \right) \\{{{\Delta\;{T_{O\; 2}(k)}} = {{Cz} \cdot {{Xz}(k)}}}{{{where}\mspace{14mu}{{Xz}(k)}} = \left( {{\Delta\;{T_{O\; 2}(k)}},{{Tht}(k)}} \right)^{T}}{{Az} = \begin{bmatrix}{1 - {{Ax} \cdot {dt}} - {{Bx} \cdot {dt}} - {{Ex} \cdot {dt}}} & {{Bx} \cdot {dt}} \\{{Cx} \cdot {dt}} & {1 - {{Cx} \cdot {dt}} - {{Fx} \cdot {dt}}}\end{bmatrix}}{{Bz} = \left( {0,{{Dx} \cdot {dt}}} \right)^{T}}{{Ez} = \left( {{{Ax} \cdot {dt}},0} \right)^{T}}{{Cz} = \left( {1,0} \right)}} & \left( {30\text{-}2} \right)\end{matrix}$

From the equations (30-1), (30-2), there is obtained the followingequation (31):

$\begin{matrix}\begin{matrix}{{\Delta\;{T_{O\; 2}(k)}} =} & {{Cz} \cdot {{Xz}\left( {k + 1} \right)}} \\{=} & {{{Cz} \cdot {Az} \cdot {{Xz}(k)}} + {{{Cz} \cdot {Bz} \cdot \Delta}\;{{SDUT}(k)}} +} \\ & {{{Cz} \cdot {Ez} \cdot \Delta}\;{{Tgd}(k)}}\end{matrix} & (31)\end{matrix}$

The equation (31) is the same as the equation (11-2), and is produced byrewriting the equation (11-2) using the matrix Az and the vectors Bz,Ez, Cz defined by the definition clauses of the equations (22-1),(22-2).

In the present embodiment, a change (difference) Δe per each given timein the element temperature difference e defined by the equation (12) isdefined by Δe(k+1)=e(k+1)−e(k). The following equation (32) is obtainedfrom this definition formula and the equation (31):

$\begin{matrix}\begin{matrix}{{e\left( {k + 1} \right)} =} & {{e(k)} + {\Delta\;{e\left( {k + 1} \right)}}} \\{=} & {{e(k)} + \left( {{\Delta\;{T_{O\; 2}\left( {k + 1} \right)}} - {\Delta\;{R\left( {k + 1} \right)}}} \right)} \\{=} & {{e(k)} + {{Cz} \cdot {Az} \cdot {{Xz}(k)}} + {{{Cz} \cdot {Bz} \cdot \Delta}\;{{SDUT}(k)}} +} \\ & {{{{Cz} \cdot {Ez} \cdot \Delta}\;{{Tgd}(k)}} - {\Delta\;{R\left( {k + 1} \right)}}}\end{matrix} & (32)\end{matrix}$

The equation (32) and the above equation (30-1) are put together,providing the following equation (33):

$\begin{matrix}{{{X\; 2\left( {k + 1} \right)} = {{\phi\;{2 \cdot X}\; 2(k)} + {G\;{2 \cdot \Delta}\;{{SDUT}(k)}} + {{Gd}\;{2 \cdot \Delta}\;{{Tgd}(k)}} + {{Gr}\;{2 \cdot \Delta}\;{R\left( {k + 1} \right)}}}}{{{where}\mspace{14mu} X\; 2(k)} = \left( {{e(k)},{\Delta\;{T_{O\; 2}(k)}},{{Tht}(k)}} \right)^{T}}{{\Phi\; 2} = \begin{bmatrix}1 & {1 - {{Ax} \cdot {dt}} - {{Bx} \cdot {dt}} - {{Ex} \cdot {dt}}} & {{Bx} \cdot {dt}} \\0 & {1 - {{Ax} \cdot {dt}} - {{Bx} \cdot {dt}} - {{Ex} \cdot {dt}}} & {{Bx} \cdot {dt}} \\0 & {{Cx} \cdot {dt}} & {1 - {{Cx} \cdot {dt}} - {{Fx} \cdot {dt}}}\end{bmatrix}}{{G\; 2} = \left( {0,0,{{Dx} \cdot {dt}}} \right)^{T}}{{{Gd}\; 2} = \left( {{{Ax} \cdot {dt}},{{Ax} \cdot {dt}},\; 0} \right)^{T}}{{{Gr}\; 2} = \left( {{- 1},0,0} \right)^{T}}} & (33)\end{matrix}$

The equation (33) is a basic formula of the model to be controlled bythe heater controller 22 according to the present embodiment. In thismodel to be controlled, the state quantity to be controlled is a statequantity vector X2(k)=(e(k), ΔT_(O2)(k), ΔTht(k))^(T) comprising theelement temperature difference e, the change ΔT_(O2) per given time inthe element temperature T_(O2), and the change ΔTht per given time inthe heater temperature Tht.

In the present embodiment, for the reasons which are the same as withthe first embodiment, the cycle time of the cycle time of the processingsequence of the heater controller 22 is longer than the period of theprocessing sequence of the element temperature observer 20 and theexhaust temperature observer 19. Consequently, the equation of the modelto be controlled which is actually used in the present embodiment is thefollowing equation (34) using the period dtc of the processing sequenceof the heater controller 22 and the ordinal number n of the processingperiod thereof:

$\begin{matrix}{{{X\; 2\left( {n + 1} \right)} = {{\phi\;{2 \cdot X}\; 2(n)} + {G\;{2 \cdot \Delta}\;{{SDUT}(n)}} + {{Gd}\;{2 \cdot \Delta}\;{{Tgd}(n)}} + {{Gr}\;{2 \cdot \Delta}\;{R\left( {n + 1} \right)}}}}{{{where}\mspace{14mu} X\; 2(n)} = \left( {{e(n)},{\Delta\;{T_{O\; 2}(n)}},{{Tht}(n)}} \right)^{T}}{{\Phi\; 2} = \begin{bmatrix}1 & {1 - {{Ax} \cdot {dtc}} - {{Bx} \cdot {dtc}} - {{Ex} \cdot {dtc}}} & {{Bx} \cdot {dtc}} \\0 & {1 - {{Ax} \cdot {dtc}} - {{Bx} \cdot {dtc}} - {{Ex} \cdot {dtc}}} & {{Bx} \cdot {dtc}} \\0 & {{Cx} \cdot {dtc}} & {1 - {{Cx} \cdot {dtc}} - {{Fx} \cdot {dtc}}}\end{bmatrix}}{{G\; 2} = \left( {0,0,{{Dx} \cdot {dtc}}} \right)^{T}}{{{Gd}\; 2} = \left( {{{Ax} \cdot {dtc}},{{Ax} \cdot {dtc}},\; 0} \right)^{T}}{{{Gr}\; 2} = \left( {{- 1},0,0} \right)^{T}}} & (34)\end{matrix}$

The basic control input SDUT (a control input at the time the batteryvoltage VB is equal to the reference value NVB) to be determined by theheater controller 22 according to the optimum predictive controlalgorithm based on the model equation (34) is given by the equation (36)shown below as having integrated ΔSDUT (a control input on the modelaccording to the equation (34)) which minimizes an evaluating functionJ2 according to the following equation (35):

$\begin{matrix}{{{J\; 2} = {\sum\limits_{n = {{- M} + 1}}^{\infty}\;\left\lbrack {{X\; 2^{T}{(n) \cdot Q}\;{0 \cdot X}\; 2(n)} + {\Delta\;{{{SDUT}^{T}(n)} \cdot H}\;{0 \cdot \Delta}\;{{SDUT}(n)}}} \right\rbrack}}{{{where}\mspace{14mu} M} = {\max\left( {{Mr},\;{Md}} \right)}}} & (35) \\{{{SDUT}(n)} = {{{Fs}\; 2} + {\sum\limits_{j = 1}^{n}\;{e(j)}} + {{Fx}\;{2 \cdot {T_{O\; 2}(n)}}} + {{Fx}\;{3 \cdot {{Tht}(n)}}} + {\sum\limits_{i = 1}^{Mr}\;\left\lbrack {{Fr}\; 2{(i) \cdot {R\left( {n + I} \right)}}} \right\rbrack} + {{Fdt}\;{2 \cdot {{Tgd}(n)}}}}} & (36)\end{matrix}$

The coefficients Fs2, Fx2, Fx3 in the first through third terms, thecoefficient Fr2(i) (i=0, 1, . . . , Mr) in the fourth term, and thecoefficient Fdt2 in the fifth term on the right side of the equation(36) are coefficients given respectively by the following equations(37-1) through (37-3):

$\begin{matrix}\begin{matrix}{{F\; 2} \equiv \left( {{{Fs}\; 2},{{Fx}\; 2},{{Fx}\; 3}} \right)} \\{= {{{- \left\lbrack {{H\; 0} + {G\;{2^{T} \cdot P}\;{2 \cdot G}\; 2}} \right\rbrack^{- 1}} \cdot G}\;{2^{T} \cdot P}\;{2 \cdot \Phi}\; 2}}\end{matrix} & \left( {37\text{-}1} \right) \\{{{{Fr}\; 2(i)} = {{{- \left\lbrack {{H\; 0} + {G\;{2^{T} \cdot P}\;{2 \cdot G}\; 2}} \right\rbrack^{- 1}} \cdot G}\;{2^{T} \cdot \left( {\zeta\; 2^{T}} \right)^{i - 1} \cdot P}\;{2 \cdot G}\; r\; 2}}\left( {{i = 1},2,\cdots,{Mr}} \right)} & \left( {37\text{-}2} \right) \\{{{{Fdt}\; 2} = {\sum\limits_{i = 0}^{Md}\;\left\{ {{{- \left\lbrack {{H\; 0} + {G\;{2^{T} \cdot P}\;{2 \cdot G}\; 2}} \right\rbrack^{- 1}} \cdot G}\;{2^{T} \cdot \left( {\zeta\; 2^{T}} \right)^{i} \cdot P}\;{2 \cdot {Gd}}\; 2} \right\}}}{{{where}\mspace{14mu} P\; 2} = {{Q\; 0} + {{{\Phi 2}^{T} \cdot P}\;{2 \cdot {\Phi 2}}} - {{{\Phi 2} \cdot P}\;{2 \cdot G}\;{2 \cdot \left\lbrack {{H\; 0} + {G\;{2^{T} \cdot P}\;{2 \cdot G}\; 2}} \right\rbrack^{- 1} \cdot G}\;{2^{T} \cdot P}\;{2 \cdot {\Phi 2}}}}}{{\zeta\; 2} = {{\Phi 2} + {G\;{2 \cdot F}\; 2}}}} & \left( {37\text{-}3} \right)\end{matrix}$

In the present embodiment, the weighted matrixes Q0, H0 with respect tothe evaluating function J2, the target value predicting time Mr, and theexhaust gas temperature predicting time Md are identical to those in thefirst embodiment. However, they may be set to values different fromthose in the first embodiment. The coefficients Fs2, Fx2, Fx3, Fr2(i),Fdt2 in the equation (36) may not necessarily be of the values accordingto the defining equations (37-1) through (37-3), but may be of valuesadjusted by way of simulation or experimentation. Furthermore, thecoefficients Fs2, Fx2, Fx3, Fr2(i), Fdt2 may be changed depending on theelement temperature, the heater temperature, etc. In the presentembodiment, as with the first embodiment, the exhaust gas temperatureTgd is maintained at the present value in the future until after Mdsteps. However, if Tgd at each time in the future can be detected orestimated, then the control input DUT may be determined using thosevalues (in this case, Fdt2 is a vector).

The above equation (36) is a formula for sequentially calculating abasic control input SDUT(n) with which the heater controller 22 controlsthe heater 13 in the present embodiment. Specifically, the heatercontroller 22 sequentially calculates the basic control input SDUT(n) ineach cycle time (period) of the control processing of the heatercontroller 22 according to the equation (36), and corrects the basiccontrol input SDUT(n) depending on the battery voltage VB according tothe equation (25), thereby determining the duty cycle DUT(n), as withthe first embodiment. The first through third terms (the term includingΣe(j) through the term including Tht(n)) on the right side of theequation (36) represent a control input component (a feedback componentbased on an optimum control algorithm) depending on the heatertemperature difference e, the element temperature T_(O2), and the heatertemperature Tht. More specifically, the first term represents a controlinput component depending on the heater temperature difference e, thesecond term represents a control input component depending on theelement temperature T_(O2), and the third term represents a controlinput component depending on the heater temperature Tht. The fourth term(the term of ΣFr2(i)·R(n+i)) on the right side of the equation (36) andthe fifth term (the term including Tgd(n)) on the right side thereofrepresent control input components (feed-forward components based on apredictive control algorithm) depending on the target value R and theexhaust gas temperature Tgd, respectively.

As the element temperature T_(O2) and the heater temperature Tht whichare required to determine the basic control input SDUT(n) according tothe equation (36), there are employed, respectively, the latest value ofthe estimated value of the element temperature T_(O2) determined by theelement temperature observer 20 and the latest value of the estimatedvalue of the heater temperature Tht. As the exhaust gas temperature Tgdin the equation (36), there is employed the latest value of theestimated value of the exhaust gas temperature Tgd determined by theexhaust temperature observer 19.

The heater temperature difference e required for the calculationaccording to the equation (36) is calculated from the latest value ofthe estimated value of the element temperature T_(O2) determined by theelement temperature observer 20 and the target value R(n) that has beenset in a cycle time before the target value predicting time Mr by thetarget value setting means 21. The fourth term on the right side of theequation (36) is calculated using time-series data R(n+1), R(n+2), . . ., R(n+Mr) of the target value R which have been set from a cycle timewhich is one cycle time after the cycle time before the target valuepredicting time Mr to the present time by the target value setting means21.

The other processing details than those described above are identical tothose according to the first embodiment. More specifically, in thepresent embodiment, the heater controller 22 determines DUT(n) accordingto the equations (36), (37) in STEP5-4 shown in FIG. 10. The otherprocessing details than the processing in STEP5-4 are exactly the sameas those according to the first embodiment. In the present embodiment,as with the first embodiment, the element temperature T_(O2) iscontrolled at the target value R. In this case, the basic control inputSDUT does not include an input component that is directly proportionalto the element temperature difference e, but includes a component (thesecond term of the equation (36)) depending on the element temperatureT_(O2), unlike the first embodiment. Therefore, immediately after theengine 1 starts to operate or when the target value R is changed by thetarget value setting means 21 from a low-temperature target value (600°C.) to a high-temperature target value (750° C. through 800° C.), as theelement temperature T_(O2) approaches the target value R, the elementtemperature T_(O2) is effectively prevented from overshooting withrespect to the target value R. The advantages produced by the basiccontrol input SDUT and the duty cycle DUT which include input componentsdepending on the heater temperature Tht, the target value R, and theexhaust gas temperature Tgd are the same as those according to the firstembodiment.

A fourth embodiment of the present invention will be described below.The present embodiment is different from the second embodiment only asto the processing of the heater controller, and those structural orfunctional parts of the fourth embodiment which are identical to thoseof the second embodiment (including those of the first embodiment whichare incorporated in the second embodiment) are denoted by identicalfigures and reference characters, and will not be described in detailbelow.

The present embodiment is different from the second embodiment only asto the processing sequence of the heater controller 32 (FIG. 14)described in the second embodiment, and determines the duty cycle DUTaccording to an optimum predictive control algorithm in the formdescribed in the third embodiment. According to the present embodiment,specifically, the state quantity of an object to be controlled by theheater controller 32 is a state quantity vector X3(n)=(e′(n),ΔT_(O2)(n), ΔTht(n))T comprising the difference (heater temperaturedifference) e′ between the heater temperature Tht and the target valueR′, a change ΔT_(O2) per given time in the element temperature T_(O2),and a change ΔTht per given time in the heater temperature Tht. That is,the state quantity of the object to be controlled uses the heatertemperature difference e′(n), rather than the heater temperaturedifference e(n) of the state quantity vector X2(n)=(e(n), ΔT_(O2)(n),ΔTht(n))T in the third embodiment. If a change Δe′ per given time in theheater temperature difference e′ is redefined as Δe′(n+1)=e′(n+1)−e′(n),then a model to be controlled is expressed by the following equation(38) according to the same concept as with the third embodiment:

$\begin{matrix}{{{{X\; 3\left( {n + 1} \right)} = {{\phi\;{3 \cdot X}\; 3(n)} + {G\;{3 \cdot \Delta}\;{{SDUT}(n)}} + {{Gd}\;{3 \cdot \Delta}\;{{Tgd}(n)}} + {{Gr}\;{3 \cdot \Delta}\;{R^{~\prime}\left( {n + 1} \right)}}}}{where}\begin{matrix}{{{X3}(n)} = \left( {{e^{\prime}(n)},{\Delta\;{T_{02}(n)}},{{Tht}(n)}} \right)^{T}} \\{{\Phi 3} = \begin{bmatrix}1 & {{Cx} \cdot {dtc}} & {1 - {{Cx} \cdot {dtc}} - {{Fx} \cdot {dtc}}} \\0 & {1 - {{Ax} \cdot {dtc}} - {{Bx} \cdot {dtc}} - {{Ex} \cdot {dtc}}} & {{Bx} \cdot {dtc}} \\0 & {{Cx} \cdot {dtc}} & {1 - {{Cx} \cdot {dtc}} - {{Fx} \cdot {dtc}}}\end{bmatrix}} \\{{G\; 3} = \left( {{{Dx} \cdot {dtc}},0,{{Dx} \cdot {dtc}}} \right)^{T}} \\{{{Gd}\; 3} = \left( {0,{{Ax} \cdot {dtc}},0} \right)^{T}} \\{{{Gr}\; 3} = \left( {{- 1},0,0} \right)^{T}}\end{matrix}}\mspace{11mu}} & (38)\end{matrix}$

The basic control input SDUT (a control input at the time the batteryvoltage VB is equal to the reference value NVB) to be determined by theheater controller 32 according to the present embodiment is given by theequation (40) shown below as having integrated ΔSDUT (a control input onthe model according to the equation (38)) which minimizes an evaluatingfunction J3 according to the following equation (39):

$\begin{matrix}\begin{matrix}{{J\; 3} = {\sum\limits_{n = {{- M} + 1}}^{\infty}\;{\begin{bmatrix}{{X\; 3^{T}{(n) \cdot Q}\;{0 \cdot X}\; 3(n)} +} \\{\Delta\;{{{SDUT}^{T}(n)} \cdot H}\;{0 \cdot \Delta}\;{{SDUT}(n)}}\end{bmatrix}\mspace{11mu}{where}}}} \\{M = {\max\left( {{Mr},{Md}} \right)}}\end{matrix} & (39) \\{{{SDUT}(n)} = {{{Fs}\; 3} + {\sum\limits_{j = 1}^{n}\;{e^{\prime}(j)}} + {{Fx}\;{4 \cdot {T_{02}(n)}}} + {{Fx}\;{5 \cdot {{Tht}(n)}}} + {\sum\limits_{i = 1}^{Mr}\;\left\lbrack {{Fr}\; 3{(i) \cdot {R^{\prime\;}\left( {n + i} \right)}}} \right\rbrack} + {{Fdt}\;{3 \cdot {{Tgd}(n)}}}}} & (40)\end{matrix}$

The coefficients Fs3, Fx4, Fx5 in the first through third terms, thecoefficient Fr3(i) (i=0, 1, . . . , Mr) in the fourth term, and thecoefficient Fdt3 in the fifth term on the right side of the equation(40) are coefficients given respectively by the following equations(41-1) through (41-3):

$\begin{matrix}\begin{matrix}{{F\; 3} \equiv \left( {{{Fs}\; 3},{{Fx}\; 4},{{Fx}\; 5}} \right)} \\{= {{{- \left\lbrack {{H\; 0} + {G\;{3^{T} \cdot P}\;{3 \cdot G}\; 3}} \right\rbrack^{- 1}} \cdot G}\;{3^{T} \cdot P}\;{3 \cdot {\Phi 3}}}}\end{matrix} & \left( {41\text{-}1} \right) \\\begin{matrix}{{{Fr}\; 3(i)} = {{{- \left\lbrack {{H\; 0} + {G\;{3^{T} \cdot P}\;{3 \cdot G}\; 3}} \right\rbrack^{- 1}} \cdot G}\;{3^{T} \cdot \left( {\zeta 3}^{T} \right)^{i - 1} \cdot P}\;{3 \cdot {Gr}}\; 3}} \\{\left( {{i = 1},2,\ldots\mspace{11mu},{Mr}} \right)}\end{matrix} & \left( {41\text{-}2} \right) \\{{{{Fdt}\; 3} = {\sum\limits_{i = 0}^{Md}\;\left\{ {{{- \left\lbrack {{H\; 0} + {G\;{3^{T} \cdot P}\;{3 \cdot G}\; 3}} \right\rbrack^{- 1}} \cdot G}\;{3^{T} \cdot \left( {\zeta 3}^{T} \right)^{i} \cdot P}\;{3 \cdot {Gd}}\; 3} \right\}}}{where}\text{}{{P\; 3} = {{Q\; 0} + {{{\Phi 3}^{T} \cdot P}\;{3 \cdot {\Phi 3}}} - {{{\Phi 3} \cdot P}\;{3 \cdot G}\;{3 \cdot \left\lbrack {{H\; 0} + {G\;{3^{T} \cdot P}\;{3 \cdot G}\; 3}} \right\rbrack^{- 1} \cdot G}\;{3^{T} \cdot P}\;{3 \cdot {\Phi 3}}}}}{{\zeta 3} = {{\Phi 3} + {G\;{3 \cdot F}\; 3}}}} & \left( {41\text{-}3} \right)\end{matrix}$

In the present embodiment, the weighted matrixes Q0, H0 with respect tothe evaluating function J3, the target value predicting time Mr, and theexhaust gas temperature predicting time Md are identical to those in thefirst embodiment. However, they may be set to values different fromthose in the first embodiment. The coefficients Fs3, Fx4, Fx5, Fr3(i),Fdt3 in the equation (40) may not necessarily be of the values accordingto the defining equations (41-1) through (41-3), but may be of valuesadjusted by way of simulation or experimentation. Furthermore, thecoefficients Fs3, Fx4, Fx5, Fr3(i), Fdt3 may be changed depending on theelement temperature, the heater temperature, etc. In the presentembodiment, as with the first embodiment, the exhaust gas temperatureTgd is maintained at the present value in the future until after Mdsteps. However, if Tgd at each time in the future can be detected orestimated, then the control input DUT may be determined using thosevalues (in this case, Fdt3 is a vector).

The above equation (40) is a formula for sequentially calculating abasic control input SDUT(n) with which the heater controller 32 controlsthe heater 13 in the present embodiment. Specifically, the heatercontroller 32 sequentially calculates the basic control input SDUT(n) ineach cycle time (period) of the control processing of the heatercontroller 32 according to the equation (40), and corrects the basiccontrol input SDUT(n) depending on the battery voltage VB according tothe equation (25), thereby determining the duty cycle DUT(n), as withthe first embodiment. The first through third terms (the term includingΣe′(j) through the term including Tht(n)) on the right side of theequation (40) represent a control input component (a feedback componentbased on an optimum control algorithm) depending on the heatertemperature difference e′, the element temperature T_(O2), and theheater temperature Tht. More specifically the first term represents acontrol input component depending on the heater temperature differencee′, the second term represents a control input component depending onthe element temperature T_(O2), and the third term represents a controlinput component depending on the heater temperature Tht. The fourth term(the term of ΣFr3(i)·R′(n+i)) on the right side of the equation (40) andthe fifth term (the term including Tgd(n)) on the right side thereofrepresent control input components (feed-forward components based on apredictive control algorithm) depending on the target value R′ and theexhaust gas temperature Tgd, respectively.

As the element temperature T_(O2) and the heater temperature Tht whichare required to determine the basic control input SDUT(n) according tothe equation (40), there are employed, respectively, the latest value ofthe estimated value of the element temperature T_(O2) determined by theelement temperature observer 20 and the latest value of the estimatedvalue of the exhaust gas temperature Tgd determined by the exhausttemperature observer 19. The heater temperature difference e′ requiredfor the calculation according to the equation (40) is calculated fromthe latest value of the estimated value of the heater temperature Thtdetermined by the element temperature observer 20 and the target valueR′(n) that has been set in a cycle time before the target valuepredicting time Mr by the target value setting means 31. The fourth termon the right side of the equation (40) is calculated using time-seriesdata R′(n+1), R′(n+2), . . . , R′(n+Mr) of the target value R′ whichhave been set from a cycle time which is one cycle time after the cycletime before the target value predicting time Mr to the present time bythe target value setting means 31.

The other processing details than those described above are identical tothose according to the second embodiment. In the present embodiment, aswith the second embodiment, the heater temperature Tht is controlled atthe target value R′, and the element temperature T_(O2) is controlled ata temperature corresponding to the target value R′. In this case, thebasic control input SDUT does not include an input component that isdirectly proportional to the element temperature difference e′, butincludes a component (the third term of the equation (40)) depending onthe heater temperature Tht, unlike the second embodiment. Therefore,immediately after the engine 1 starts to operate or when the targetvalue R′ is changed by the target value setting means 31 from alow-temperature target value (700° C. in the present embodiment) to ahigh-temperature target value (850° C. through 900° C.), as the heatertemperature Tht approaches the target value R′, the heater temperatureTht is effectively prevented from overshooting with respect to thetarget value R′. The element temperature T_(O2) is thus allowed toconverge smoothly to a temperature corresponding to the target value R′for the heater temperature Tht. The advantages produced by the basiccontrol input SDUT and the duty cycle DUT which include input componentsdepending on the heater temperature T_(O2), the target value R′, and theexhaust gas temperature Tgd are the same as those according to thesecond embodiment.

In the first through fourth embodiments described above, the exhaust gastemperature Tgd is estimated. However, an exhaust gas sensor may bedisposed in the vicinity of the O₂ sensor 8, and the exhaust gastemperature Tgd may be detected by the exhaust gas sensor. Even if theexhaust gas sensor is disposed in a location remote from the O₂ sensor8, if the temperature detected by the exhaust gas sensor issubstantially the same as the exhaust gas temperature Tgd near the O₂sensor 8 due to the layout of the exhaust system or the like, then thedetected temperature may be substituted for the exhaust gas temperatureTgd near the O₂ sensor 8. If the temperature detected by the exhaust gassensor is used, then the temperature (the latest value) detected by theexhaust gas sensor is used as the value of the exhaust gas temperatureTgd in the equation (10-1) to estimate the element temperature T_(O2)and the heater temperature Tht. Furthermore, the temperature (the latestvalue) detected by the exhaust gas sensor is used as the value of theexhaust gas temperature Tgd in the equation (24), the equation (28), theequation (36), or the equation (40) to calculate the basic duty cycleSDUT and then calculate the duty cycle DUT according to the equation(25).

If the exhaust gas sensor is disposed in a location remote from the O₂sensor 8 and if the temperature detected by the exhaust gas sensor isnot necessarily be same as the exhaust gas temperature Tgd near the O₂sensor 8, then it is possible to estimate the exhaust gas temperatureTgd near the O₂ sensor 8 using the temperature detected by the exhaustgas sensor. For example, if an exhaust gas temperature sensor isprovided which is capable of detecting the exhaust gas temperature Tgdin the partial exhaust passageway 3 b, then the exhaust gas temperatureTgd near the O₂ sensor 8 can be estimated by the calculations of theequations (8), (9) by using the detected temperature as the exhaust gastemperature Tgb in the equation (8-1). In this case, the calculations ofthe equations (1), (5), (6) are not required.

In the first and second embodiments, both the element temperature T_(O2)and the heater temperature Tht are estimated. However, either one ofthem may be detected directly by a temperature sensor. If the elementtemperature T_(O2) is detected, then the heater temperature Tht may beestimated using the detected value of the element temperature T_(O2) asthe value of the element temperature T_(O2) in the equation (10-2). Theduty cycle DUT may be calculated using the estimated value as the heatertemperature Tht in the calculation of the equation (24), the equation(28), the equation (36), or the equation (40), and also using thedetected value of the element temperature T_(O2) as the value of theelement temperature T_(O2). If the heater temperature Tht is detected,then the element temperature T_(O2) may be estimated using the detectedvalue of the heater temperature Tht as the value of the heatertemperature Tht in the equation (10-1). The duty cycle DUT may becalculated using the estimated value as the element temperature T_(O2)in the calculation of the equation (24), the equation (28), the equation(36), or the equation (40), and also using the detected value of theheater temperature Tht as the value of the heater temperature Tht. Ifboth the element temperature T_(O2) and the heater temperature Tht aredetected by the temperature sensors, then the detected temperatures maybe used as the values of the element temperature T_(O2) and the heatertemperature Tht in the calculation of the equation (24), the equation(28), the equation (36), or the equation (40) to calculate the dutycycle DUT.

In the first and second embodiments, the element temperature T_(O2) ofthe O₂ sensor 8 or the heater temperature Tht is controlled at thetarget value R or R′ according to the optimum predictive controlalgorithm. However, the present invention is not limited to the optimumpredictive control algorithm.

For example, the control input DUT may be determined according to anordinary optimum control algorithm which includes no predictive controlalgorithm. In this case, the control input DUT may sequentially becalculated according to an equation which is produced by removing thefourth term (the term including R(n+i)) and the fifth term (the termincluding Tgd(n)) from the right side of the equation (24) or theequation (36) or by removing the fourth term (the term includingR′(n+i)) and the fifth term (the term including Tgd(n)) from the rightside of the equation (28) or the equation (40). According to thismodification, the heater controller for determining the control inputDUT is an optimum servocontroller for determining the control input DUTin order to minimize the value of the evaluating function J0 or J1 or J2or J3 where M=0 in the equation (17), the equation (27), the equation(35) or the equation (39).

Alternatively, the control input DUT may calculated according to anequation which is produced by removing one or two of the third term (thecomponent depending on the heater temperature Tht), the fourth term (theterm including R(n+i)), and the fifth term (the term including Tgd(n))from the right side of the equation (24). Further alternatively, thecontrol input DUT may calculated according to an equation which isproduced by removing one or two of the third term (the componentdepending on the element temperature T_(O2)), the fourth term (the termincluding R′(n+i)), and the fifth term (the term including Tgd(n)) fromthe right side of the equation (28). Further alternatively, the controlinput DUT may calculated according to an equation which is produced byremoving one or two of the third term (the component depending on theheater temperature Tht), the fourth term (the term including R(n+i)),and the fifth term (the term including Tgd(n)) from the right side ofthe equation (36). Further alternatively, the control input DUT maycalculated according to an equation which is produced by removing one ortwo of the second term (the component depending on the elementtemperature T_(O2)), the fourth term (the term including R′(n+i)), andthe fifth term (the term including Tgd(n)) from the right side of theequation (40). The component depending on the element temperaturedifference e in the equation (24) or the equation (36) and the componentdepending on the heater temperature difference e′ in the equation (28)or the equation (40) may be determined according to a PI control law ora PID control law.

In each of the embodiments described above, the element temperatureT_(O2) of the O₂ sensor 8 is controlled. However, the present inventionis also applicable to an exhaust gas sensor other than the O₂ sensor 8(e.g., the wide-range air-fuel ratio sensor 9 or a humidity sensor forgenerating an output signal representative of the water content of theexhaust gas).

The internal combustion engine to which the present invention isapplicable may be an ordinary port-injected internal combustion engine,a spark-ignition internal combustion engine with direct fuel injectioninto cylinders, a diesel engine, an internal combustion engine for useas an outboard engine on a boat, etc.

INDUSTRIAL APPLICABILITY

As described above, the present invention is useful as a technology forappropriately controlling the temperature of an exhaust gas sensordisposed in the exhaust system of an internal combustion engine mountedon an automobile, a hybrid vehicle, an outboard engine assembly, or thelike, at a desirable temperature for stabilizing the outputcharacteristics of the exhaust gas sensor.

1. An apparatus for controlling a temperature of an exhaust gas sensordisposed in an exhaust passage of an internal combustion engine andhaving an active element for contacting an exhaust gas flowing throughthe exhaust passage and a heater for heating the active element,characterized by comprising: means for sequentially acquiring elementtemperature data representing the temperature of said active element,means for sequentially acquiring heater temperature data representingthe temperature of said heater, and heater control means forsequentially generating a control input which defines an amount of heatgenerating energy supplied to said heater so as to equalize thetemperature of the active element represented by said elementtemperature data to a predetermined target temperature, and controllingthe heater depending on the control input, and characterized in thatsaid control input generated by said heater control means includes atleast an input component depending on a difference between thetemperature of the active element represented by said elementtemperature data and said target temperature and an input componentdepending on the temperature of the heater represented by said heatertemperature data.
 2. An apparatus for controlling the temperature of anexhaust gas sensor according to claim 1, characterized by comprisingmeans for sequentially acquiring exhaust gas temperature datarepresenting the temperature of said exhaust gas, and characterized inthat said control input generated by said heater control means includesan input component depending on the temperature of the exhaust gasrepresented by said exhaust gas temperature data.
 3. An apparatus forcontrolling the temperature of an exhaust gas sensor according to claim1, characterized in that said control input generated by said heatercontrol means includes an input component depending on said targettemperature.
 4. An apparatus for controlling the temperature of anexhaust gas sensor according to claim 1, characterized in that saidcontrol input generated by said heater control means includes an inputcomponent depending on the temperature of the active element representedby said element temperature data.
 5. A method of controlling atemperature of an exhaust gas sensor disposed in an exhaust passage ofan internal combustion engine and having an active element forcontacting an exhaust gas flowing through the exhaust passage and aheater for heating the active element, characterized by comprising thesteps of: sequentially acquiring element temperature data representingthe temperature of said active element and heater temperature datarepresenting the temperature of said heater, sequentially generating acontrol input which defines an amount of heat generating energy suppliedto said heater so as to equalize the temperature of the active elementrepresented by said element temperature data to a predetermined targettemperature, and controlling the heater depending on the control input,and characterized in that when said control input is generated, saidcontrol input is generated so as to include at least an input componentdepending on a difference between the temperature of the active elementrepresented by said element temperature data and said target temperatureand an input component depending on the temperature of the heaterrepresented by said heater temperature data.
 6. A method of controllingthe temperature of an exhaust gas sensor according to claim 5, furthercharacterized by comprising the step of sequentially acquiring exhaustgas temperature data representing the temperature of said exhaust gas,and characterized in that when said control input is generated, saidcontrol input is generated so as to further include an input componentdepending on the temperature of the exhaust gas represented by saidexhaust gas temperature data.
 7. A method of controlling the temperatureof an exhaust gas sensor according to claim 5, characterized in thatwhen said control input is generated, said control input is generated soas to further include an input component depending on said targettemperature.
 8. A method of controlling the temperature of an exhaustgas sensor according to claim 5, characterized in that when said controlinput is generated, said control input is generated so as to furtherinclude an input component depending on the temperature of the activeelement represented by said element temperature data.
 9. A recordingmedium readable by a computer and storing a temperature control programfor enabling the computer to perform a process of controlling atemperature of an active element of an exhaust gas sensor disposed in anexhaust passage of an internal combustion engine and having the activeelement for contacting an exhaust gas flowing through the exhaustpassage and a heater for heating the active element, characterized inthat said temperature control program includes a program for enablingsaid computer to perform a process of sequentially acquiring elementtemperature data representing the temperature of said active element andheater temperature data representing the temperature of said heater, acontrol input generating program for enabling said computer to perform aprocess of sequentially generating a control input which defines anamount of heat generating energy supplied to said heater so as toequalize the temperature of the active element represented by saidelement temperature data to a predetermined target temperature, and aprogram for enabling said computer to perform a process of controllingthe heater depending on the control input, wherein said control inputgenerating program has an algorithm for enabling said computer togenerate said control input so as to include at least an input componentdepending on a difference between the temperature of the active elementrepresented by said element temperature data and said target temperatureand an input component depending on the temperature of the heaterrepresented by said heater temperature data.
 10. A recording mediumstoring a temperature control program for an exhaust gas sensoraccording to claim 9, characterized in that said temperature controlprogram further includes a program for enabling said computer to performa process of sequentially acquiring exhaust gas temperature datarepresenting the temperature of said exhaust gas, wherein said controlinput generating program has an algorithm for enabling said computer togenerate said control input so as to further include an input componentdepending on the temperature of the exhaust gas represented by saidexhaust gas temperature data.
 11. A recording medium storing atemperature control program for an exhaust gas sensor according to claim9, characterized in that said control input generating program has analgorithm for enabling said computer to generate said control input soas to further include an input component depending on said targettemperature.
 12. A recording medium storing a temperature controlprogram for an exhaust gas sensor according to claim 9, characterized inthat said control input generating program has an algorithm for enablingsaid computer to generate said control input so as to further include aninput component depending on the temperature of the active elementrepresented by said element temperature data.